Sparse-grid quadrature nonlinear filtering

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.

In this paper, vision-based relative navigation of two spacecraft is addressed using the sparse Gauss-Hermite quadrature filter. The relative navigation provides the estimation of the relative attitude and relative orbit between spacecraft, which is a challenging filtering problem since it is a highly nonlinear and high dimensional estimation problem. Many filters have been used to solve this problem, such as, the extended Kalman filter (EKF) and the unscented Kalman filter (UKF). However, these filters are not accurate enough when the initial uncertainty is large or the nonlinearity is significant. Moreover, although the conventional Gauss-Hermite quadrature filter is more accurate than the UKF, it cannot be used since a huge number of points is required. It is shown in this paper that the sparse Gauss-Hermite quadrature filter can achieve better estimation accuracy than the EKF, the UKF, and the cubature Kalman filter without excessive computation load.

Since measurements of process variables are subject to measurements errors as well as process variability, data reconciliation is the procedure of optimally adjusting measured date so that the adjusted values obey the conservation laws and constraints. Thus, data reconciliation for dynamic systems is fundamental and important for control, fault detection, and system optimization. Attempts to successfully implement estimators are often hindered by serve process nonlinearities, complicated state constraints, and un-measurable perturbations. As a constrained minimization problem, the dynamic data reconciliation is dynamically carried out to product smoothed estimates with variances from the original data. Many algorithms are proposed to solve such state estimation such as the extended Kalman filter (EKF), the unscented Kalman filter, and the cubature Kalman filter (CKF). In this paper, we investigate the use of CKF algorithm in comparative with the EKF to solve the nonlinear dynamic data reconciliation problem. First we give a broad overview of the recursive nonlinear data dynamic reconciliation (RNDDR) scheme, then present an extension to the CKF algorithm, and finally address the issue of how to solve the constraints in the CKF approach. The CCRNDDR method is proposed by applying the RNDDR in the CKF algorithm to handle nonlinearity and algebraic constraints and bounds. As the sampling idea is incorporated into the RNDDR framework, more accurate estimates can obtained via the recursive nature of the estimation procedure. The performance of the CKF approach is compared with EKF and RNDDR on nonlinear process systems with constraints. The conclusion is that with an error optimization solution of the correction step, the reformulated CKF shows high performance on the selection of nonlinear constrained process systems. Simulation results show the CCRNDDR is an efficient, accurate and stable method for real-time state estimation for nonlinear dynamic processes.

In this paper, the state trajectories of a hypersonic vehicle is estimated using a new generalized cubature Kalman filter. This new nonlinear estimation technique is based on a class of arbitrary-degree cubature rules to compute the multivariate integrals in the Bayesian filtering algorithm and can achieve higher degree of estimation accuracy. The performance of this new cubature Kalman filter is compared with the Extended Kalman filter, the unscented Kalman filter and the conventional third-degree cubature Kalman filter. The simulation results demonstrate that the proposed filter outperforms those three filters in estimation accuracy and convergence rate. It is also computationally more efficient than many other point-based estimation techniques.

In this paper, the Gauss-Hermite quadrature filter (GHQF) is introduced to estimate battery state of charge (SOC) based on a common electrical analogue battery model useful for realtime applications. A high fidelity circuit-based electrical analogue battery model was used with a new parameter estimation algorithm. When the model runs off-line, the terminal voltage estimation error is at the 30mV level for a battery module with 14.4V nominal voltage, which is more accurate than previously reported comparable behavioral models. However, the battery SOC is calculated based on coulomb-counting, which is considered inaccurate and has many drawbacks, such as the problem of estimating the initial SOC and the current loss problem. In this paper a filter is used to improve the SOC estimation accuracy for the battery model to be used for realtime applications. The extended Kalman filter (EKF), the unscented Kalman filter (UKF) and the Gauss-Hermite quadrature filter are all studied and compared in this work. The Gauss-Hermite quadrature filter excels among the commonly used filters when used with the high fidelity electrical analogue battery model. This conclusion is experimentally verified on a 6.8 Ah Lithium-ion battery module.

Due to the uncertainty of the Radio Frequency Identification (RFID) measurements and limit of the placement of the readers, it's necessary to use the estimation method to obtain more accurate trajectory in RFID indoor tracking system. The traditional recursive estimation from K to K+1 sampling point may fail, because the measurement of RFID system is irregular sampling due to the data-driven measurement mechanism. This paper develops the tracking method for indoor RFID system, including estimation dynamic model based on the estimated states and nonlinear fusion estimation algorithm for variable-irregular sampling measurements. Two estimation methods are given based on the Extended Kalman filter (EKF) and Unscented Kalman filter (UKF), respectively. The tracking performances are compared and the simulation results show that the performance of UKF can get better performance for indoor RFID tracking, especially in the low detection rate area.

The trajectory estimation of a highly maneuvering target is a challenging problem and has practical applications. The interacting multiple model (IMM) filter is a well-established filtering algorithm for the trajectory estimation of maneuvering targets. In this study, we present an overview of IMM filtering algorithms for tracking a highly-maneuverable fighter aircraft using an air moving target indicator (AMTI) radar on another aircraft. This problem is a nonlinear filtering problem due to nonlinearities in the dynamic and measurement models. We first describe single-model nonlinear filtering algorithms: the extended Kalman filter (EKF), unscented Kalman filter (UKF), and cubature Kalman filter (CKF). Then, we summarize the IMM-based EKF (IMM-EKF), IMM-based UKF (IMM-UKF), and IMM-based CKF (CKF). In order to compare the state estimation accuracies of the IMM-based filters, we present a derivation of the posterior Cramér-Rao lower bound (PCRLB). We consider fighter aircraft traveling with accelerations 3g, 4g, 5g, and 6g and present numerical results for state estimation accuracy and computational cost under various operating conditions. Our results show that under normal operating conditions, the three IMM-based filters have nearly the same accuracy. This is due to the accuracy of the measurements of the AMTI radar and the high data rate.

Performance evaluation of Cubature Kalman filter in a GPS/IMU tightly-coupled navigation system

According to the problem on tracking planning of mobility assistance robot, a tracking planning method was presented using the kalman filter (KF). Based on known nonholonomic dynamic model of mobility assistance robot, the cubature kalman filter (CKF) was used to improve the influence of nonlinear system, and the autonomous navigation and control of mobility assistance robot was realized. The experimental results indicate that the method presented in this paper was excellent than extended kalman filter (EKF) and unscented kalman filter (UKF) in the stability, precision and computational complexity.

RADAR plays a crucial role in target tracking. A faulty tracking is vulnerable to fatal errors. Since RADAR signals contain noise in them, filters are used to remove the noise. This paper implements Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF) and Cubature Kalman Filter (CKF) for radar target tracking. Each filter is exposed to the same environment and the results are observed. A radar target tracking model is proposed with an input chirp signal of 77 GHz. The turn rate is varied between 0.2 and 150. The simulations are carried out in MATLAB. Since every simulation generates random motions of the target, the filter can be gauged only on the basis of how well the estimated trajectory follows the true trajectory. The findings have shown that EKF works best for large turn rate and high noise, in comparison to UKF and CKF, thus making it an appropriate choice for such applications.

Effect of Noises on the Performance of Nonlinear State Estimator for Semi Batch Autocatalytic Esterification Reactor

Requirements of highly precise pointing performance have been imposed on recently developed spacecrafts for a variety of missions. The stringent requirements have called on on-orbit estimation of spacecraft dynamics parameters and calibration of on-board sensors as indispensible practices. Consequently, on-orbit estimation of the mass moment of inertia of spacecraft has been a major issue mostly due to the changes by solar panel deployment and a large portion of fuel consumption (Creamer et al., 1996; Ahmed et al., 1998; Bordany et al., 2000; VanDyke et al., 2004; Myung et al., 2007; Myung & Bang, 2008; Sekhavat et al., 2009). As for measurement sensors, on-board calibration of alignment and bias errors of attitude and rate sensors is one of the main concerns of attitude sensor calibration researches (Pittelkau, 2001 & 2002, Lai et al., 2003). Pittelkau (2002) proposed an attitude estimator based on the Kalman filter (Kalman, 1960), in which spacecraft attitude quaternion, rate sensor misalignment and bias, and star tracker misalignments are taken into consideration as states, whereas the body rate is dealt as a synthesized signal by the estimates. Lai at al. (2003) derived a method for alignment estimation of attitude and rate sensors based on the unscented Kalman filter (UKF) (Julier and Uhlmann, 1997). Ma and Jiang (2005) presented spacecraft attitude estimation and calibration based only magnetometer measurements using an UKF. An interesting point is that we need predesigned 3-axis excitation manoeuvres of spacecraft for both dynamics parameter estimation and sensor calibration. Therefore, this study is motivated to merge above estimation and calibration processes into a single filtering problem. It is noteworthy that poor information of moments of inertia is to be treated as a system uncertainty while the rate sensor model errors are to be incorporated into the measurement process. As a filtering algorithm, this study employs a UKF. Extended Kalman filters (EKFs) have been successfully applied to the nonlinear attitude estimation problem (Crassidis et al., 2007). Hybrid estimation using the EKF has been reported by Myung at al. (2007). However, the EKF estimates using the first order linearization, which may lead to instability of the filter (ValDyke et al., 2004). The UKF approximates the nonlinear model to the second order by spreading points 1 sigma apart from the a priori mean. Performing nonlinear

This chapter deals with the problem of non-linear and non-Gaussian states and parameters estimation using Bayesian methods. The performances of various conventional and state-of-the-art state estimation techniques are compared when they are utilized to achieve this objective. These techniques include the Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF), and Particle Filter (PF). In the current work, the authors consider two systems (biological model and power system) to perform evaluation of estimation algorithms. The results of the comparative studies show that the UKF provides a higher accuracy than the EKF due to the limited ability of EKF to accurately estimate the mean and covariance matrix of the estimated states through lineralization of the nonlinear process model. The results also show that the PF provides a significant improvement over the UKF because, unlike UKF, PF is not restricted by linear-Gaussian assumptions which greatly extends the range of problems that can be tackled.

Composite weighted average consensus filtering for space object tracking

In this paper, a composite weighted average consensus filtering algorithm (CWACF) is proposed by applying multiple heterogeneous nonlinear filters. In light of the sensors’ different sensing accuracy and computational capability, extended Kalman filter (EKF) and sparse-grid quadrature filter (SGQF) are compositely adopted on different sensors as local filters. Then, estimation from neighbours are fused based on the weighted average consensus framework to attain better estimation performance. Moreover, it has been proved that the estimation error is exponentially bounded in mean square. The performance of the proposed algorithm and distributed extended Kalman filtering (DEKF) are compared by a target localization case through a sensor network.