Nonlinear filtering using the wavelet transform

Dynamical systems encountered in reality are essentially nonlinear. A number of approaches were propped for nonlinear filter with different accuracy. They are mainly investigated from the Bayesian point of view, and may be classified into three kinds of methods: linearization, statistical approximation and Monte-Carlo simulation (Jazwinski 1970; Tanizaki 1993; Gelb 1994). Very often, linearization of the nonlinear system is done using either a precomputed nominal trajectory or the estimate of the state vector. This second linearization approach is better known as the extended Kalman filter (Jazwinski 1970). Since the solution based on one-step linearization may be poor for a highly nonlinear system, iteration in this case is expected in order to obtain a more accurate estimate. Higher order approaches should probably be taken into account. It should be noted, however, that if the system presents a significant nonlineaaity, even the mean and covariance matrix of the nonlinear filter can be misleading, since the means of the estimated state variables may deviate appreciably from their true parameter values. The basic idea of statistical approximation is to replace a nonlinear function of random variables by a series expansion (Gelb 1994) or to approximate thea posteriori conditional probability density function of the state vector (Sorenson & Stubberud 1968; Kramer & Sorenson 1988). The Monte Carlo simulation technique may be used to determine the mean and covariance matrix of the nonlinear filter (if properly designed), which requires a large sample to obtain statistically meaningful results (see e.g. Brown & Mariano 1989; Carlin et al. 1992; Gelb 1994; Tanizaki 1993)

Object-Oriented Environment for Assessing Tracking Algorithms

LIDAR (Laser Integrated Radar) is an engineering problem of great practical importance in environmental monitoring sciences. Signal processing for LIDAR applications involves highly nonlinear models and consequently nonlinear filtering. Optimal nonlinear filters, however, are practically unrealizable. In this paper, the Lainiotis′s multi‐model partitioning methodology and the related approximate but effective nonlinear filtering algorithms are reviewed and applied to LIDAR signal processing. Extensive simulation and performance evaluation of the multi‐model partitioning approach and its application to LIDAR signal processing shows that the nonlinear partitioning methods are very effective and significantly superior to the nonlinear extended Kalman filter (EKF), which has been the standard nonlinear filter in past engineering applications.

본 논문은 비가우시안 노이즈가 존재하는 수중환경에서 비선형 필터 기법에 따른 2차원 위치 추정에 관한 연구 내용이다. 최근 위치 추정을 위한 필터로 확장형 칼만필터(EKF: Extended Kalman filter)가 많이 사용되고 있다. 하지만, 수중과 같은 비가우시안 노이즈가 존재하는 비선형 시스템에서는 많은 문제점을 가지고 있다. 따라서 본 논문에서는 상태변이의 예측을 기반으로한 EKF를 대신하여 통계적 발생인자 에 기반을 둔 분포 재해석 기법을 이용한 2차원 파티클필터 (TDPF: Two-Dimension Particle Filter)를 제안한다. 모의 실험을 통하여 Non-Gaussian Noise 가 존재하는 수중환경에서 제안하는 TDPF의 성능을 EKF와 비교분석하였으며 TDPF가 EKF보다 정확한 위치 추정결과를 제공하는 것을 확인하였다. This paper has considered the location estimation problem in two dimension space by using a non-linear filter under non-Gaussian noise in underwater acoustic sensor networks(UASNs). Recently, the extended Kalman filter (EKF) is widely used in location estimation. However, the EKF has a lot of problems in the non-linear system under the non-gaussian noise environment like underwater environment. In this paper, we propose the improved Two-Dimension Particle Filter (TDPF) using the re-interpretation distribution techniques based on the maximum likelihood (ML). Through the simulation, we compared and analyzed the proposed TDPF with the EKF under the non-Gaussian underwater sensor networks. Finally, we determined that the TDPF's result shows more accurate localization than EKF's result.

The focus of this work is the extension of nonlinear state estimation methods to gas-lifted systems. The extended Kalman filter (EKF), unscented Kalman filter (UKF) and particle filter (PF) were used to estimate the nonlinear states. Brief descriptions of the filters were first presented starting from the linear Kalman filter. Hypothesis tests on the expectation of the residuals were performed to show how close to optimal the estimation methods are and it showed the UKF estimates to be slightly better than EKF while PF performs the worst. The PF has poor accuracy using residual visualisation, hypothesis test and the root mean squared error (RMSE) values of the residuals. The gas-lifted system exhibits casing heading instability where the states show oscillatory behaviour depending on the value of the input but the results here do not change in a known way for each filter as the input is changed from the non-oscillatory region to the oscillatory region. Therefore, for this noise distribution and model assumption, either the EKF or UKF can be used for nonlinear state estimation with UKF better preferred if computational cost is not considered when control solutions are used in gas-lifted system.

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

A new nonlinear filter is derived for continuous-time processes with discrete-time measurements. The filter is exact, and it can be implemented in real time with a computational complexity that is comparable to the Kalman filter. This new filter includes both the Kalman filter and the discrete-time version of the Benes filter as special cases. Moreover, the new theory can handle a large class of nonlinear estimation problems that cannot be solved using the Kalman or discrete-time Benes filters. A simple approximation technique is suggested for practical applications in which the dynamics do not satisfy the required conditions exactly. This approximation is analogous to the so-called "extended Kalman filter" [10], and it represents a generalization of the standard linearization method.

There are a kind of non-linear filters in the active detection problem solution which can weaken the bigger and strengthen the smaller samples, that can be called as Gaussianization, and to improve performance of subsequent correlation test. An explicit definition about this kind of nonlinear Gaussianization filter is given at first. In succession, the two typical nonlinear filters are proposed and studied. One is U-filter, based on the probability density function and its derivate. The other is G-filter, based on the cumulative distribution function and its inverse. Instances with lake trial data are illustrated to test these two methods' performance.

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.

Tracking a ballistic target in its reentry mode by considering the radar measurements is a highly complex problem in nonlinear filtering. Kalman Filter (KF) is used to estimate the position of target when the measurements are corrupted with noise. If the measurements are nonlinear (radar measurements) then Extended kalman filter (EKF) is used. For obtaining reliable estimate of the target state, filter has to be tuned before the operation which is offline. Tuning an EKF is the process of estimating the process noise covariance matrix (Q) and measurement noise covariance matrix (R). This paper presents a new method of tuning the EKF using different evolutionary algorithms.

Presents the results obtained in our research about application of modern nonlinear filtering techniques to GPS based position estimation. The stand-alone GPS based position estimation problem using GPS pseudo-range and Doppler shifts measurements are described. A model for position and velocity estimation are developed. The model is nonlinear and has variable measurement number for coping with an arbitrary number of satellites. Over the last 20-30 years, the extended Kalman filter (EKF) has become the algorithm of choice in numerous nonlinear estimation and machine learning applications. In this work the use of an alternative filter: the unscented Kalman filter (UKF) is proposed. The first experimental results that comprise the comparison of estimation results obtained with a simple model using different filters are then presented. Future research directions are also discussed.

A general Bayesian nonlinear estimation method using resampled Smooth Particle Hydrodynamics solutions of the underlying Fokker–Planck Equation

In radar tracking the target motion is best modeled in Cartesian coordinates. Its position is however measured in polar coordinates (range and azimuth). Tracking in Cartesian coordinates with noisy polar measurements requires either converting the measurements to a Cartesian frame of reference and then applying the linear Kalman filter to the converted measurement or using the extended Kalman filter (EKF) in mixed coordinates. The first approach is accurate only for moderate cross-range errors; the second approach is consistent only for small errors. A new efficient tracking algorithm using the multidimensional Gauss-Hermite quadratures to propagate the mean and the covariance of the conditional probability density function is presented. This method is compared with the EKF and the converted measurement Kalman filter (CMKF) and it is shown to be more accurate.

The stochastic nature of noise signals affects the vehicle’s internal states and the outputs, resulting in the difficulty in estimation. The unknown or time-varying nature of noise signals if not taken into account for estimation, the results will diverge and be highly deteriorated. In this paper, a maximum likelihood principle (MLP) based adaptive robust extended kalman filter for estimating the states of the adopted non-linear vehicle model is proposed. An observability test is done for the purpose of estimation. A covariance matching (CM) based robust adaptive high forgetting factor based fault tolerant technique is also employed on the robust adaptive extended and unscented kalman filters for comparison purpose. The Robustness of the filter is analyzed by varying the parameter of the vehicle through a local sensitivity analysis. The results show that the MLP based approach to the extended kalman filter performs well in three simulations for sinusoidal steering, Double Lance Change, J-Turn, Fishhook, Slalom maneuver in comparison to robust adaptive unscented kalman filter. Friction coefficient of 0.8 (dry road) and 0.4 (wet road) is chosen for the simulation. The sideslip angle RMSE value for MLP based estimation is obtained as 2.62e-05, 4.545e-06 for Sine and DLC maneuver.

The paper presents a method of computational complexity reduction in Extended Kalman Filters dedicated for systems with non-linear measurement models. Extended Kalman filters are commonly used in radio-location and radio-navigation for estimating an object’s position and other parameters of motion, based on measurements, which are non-linearly related to the object’s position. This non-linearity forces designers to use non-linear filters, such as the Extended Kalman Filter mentioned, where linearization of the system’s model is performed in every run of the filter’s loop. The linearization, consisting of calculating Jacobian matrices for non-linear functions in the dynamics and/or observation models, significantly increases the number of operations in comparison to the linear Kalman filter. The method proposed in this paper consists of analyzing a variability of Jacobians and performing the model linearization only when expected changes of those Jacobians exceed a preset threshold. With a properly chosen threshold value, the proposed filter modification leads to a significant reduction of its computational burden and does not noticeably increase its estimation errors. The paper describes a practical simulation-based method of determining the threshold. The accuracy of the filter for various threshold values was tested for simplified models of radar systems.