Target Location Using the Extended Kalman Filter

Large-eddy simulations (LES) are an important tool in studying meteorological processes on the local scale. However, specific techniques are needed to generate turbulence when initializing these models. While random perturbations in the initial condition might be sufficient for periodic boundaries, their effect will rapidly diminish when performing simulations with open boundaries, which are needed to analyze undisturbed flow over isolated topographic features. Hence, more elaborate techniques are needed to generate and maintain turbulence for such boundaries.Eddy recycling is one of those turbulent inflow generation techniques and already implemented in atmospheric LES codes such as CM1 and PALM. The eddy recycling method captures the turbulent structures a certain distance downstream of the inflow and injects the captured turbulent field in the inflow region. While eddy recycling successfully generates turbulence in the inflow region, flux profiles are very sensitive to the distance between capture region and injection region as well as the geometry of these regions. In addition, artificial periodicity in the fluxes might be introduced when placing injection region and capture region too close together. The sensitivity of turbulent flux profiles towards the shape of the capture region, the specification of the turbulent field and the effect of artificial periodicity of these fluxes has so far received little attention in atmospheric LES.To address this issue and investigate the sensitivity of the profile shape of turbulent moments to the specification of the eddy recycling module, a set of idealized LES were conducted for a shear and buoyancy driven atmospheric boundary layer with the Cloud Model 1 (CM1). Each simulation had an identical inflow profile but differed in the shape of the capture region and the distance between the capture and injection region. Preliminary results show that the shape of TKE and variances of wind components as well as temperature is highly sensitive to the shape and location of both capture and injection regions, while average profiles are less affected.

Object-Oriented Environment for Assessing Tracking Algorithms

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.

In the case of a non-linear system, the dynamic state of the targets (position, velocity, and acceleration) is estimated by an extended Kalman filter (EKF). The theory of EKF is established on the assumption that measurements follow Gaussian distribution. However, in practice, this assumption falls short and limits the application of EKF. In literature, to deal with the non-Gaussianity, the maximum correntropy criterion (MCC)-based EKF (EKF-MCC) has been studied well. The MCC, an information-theoretic criterion, claims to effectively deal with the system's non-Gaussianity. Nevertheless, like EKF, EKF-MCC also approximates the known system non-linearity with a Jacobian. The Jacobian provides the first-order approximation of the non-linearity and hinders the estimation accuracy achieved by EKF-MCC, particularly for complex target motion models. Therefore, in this work, firstly, we propose to use EKF-MCC for estimating the dynamic state of the target from non-Gaussian measurement. After that, utilizing MCC, we propose reproducing kernel Hilbert space (RKHS) based non-linear estimation of system non-linearity and using it with EKF-MCC. Amid non-linear estimation utilizing MCC, the proposed filter is named EKF-MCC-RKHS. The simulation performed to estimate the dynamic states of the complex constant acceleration (CA) target motion model validates the superiority of EKF-MCC-RKHS over recently introduced EKF-MCC and traditional EKF.

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.

The Kalman filter (KF) is the optimal estimator that minimizes the mean square error when the state and measurement dynamics are linear in nature, provided the process and measurement noise processes are modeled as white Gaussian. However, in the real world, one encounters a large number of scenarios where either the process or measurement model (or both) are nonlinear. In such cases a class of suboptimal Kalman filter implementations called extended Kalman filters (EKF) are used. EKFs operate by linearizing the nonlinear model around the current reference trajectory and then designing the Kalman filter gain for the linearized model. Recently, an alternative approach has emerged for a certain class of problems where the error in the states is estimated using a Kalman filter, rather than the state itself. This error state KF (ErKF) approach, by deriving the error state dynamics, via the perturbation of the nonlinear plant, lends itself to optimal updates in the error states and optimal prediction and updates in the error state covariance. This is because the error state dynamics are linear, thereby satisfying a condition for optimal Kalman filtering. This paper offers a comparison between the EKF and ErKF via simulations and shows that the ErKF performance is robust to a variety of aircraft maneuvers performed. Furthermore, this paper shows that the ErKF, unlike the EKF, need not be repeatedly tuned with respect to the noise covariances in order to obtain acceptable estimation performance.

As one of the most essential technologies, wireless sensor networks (WSNs) integrate sensor technology, embedded computing technology, and modern network and communication technology, which have become research hotspots in recent years. The localization technique, one of the key techniques for WSN research, determines the application prospects of WSNs to a great extent. The positioning errors of wireless sensor networks are mainly caused by the non-line of sight (NLOS) propagation, occurring in complicated channel environments such as the indoor conditions. Traditional techniques such as the extended Kalman filter (EKF) perform unsatisfactorily in the case of NLOS. In contrast, the robust extended Kalman filter (REKF) acquires accurate position estimates by applying the robust techniques to the EKF in NLOS environments while losing efficiency in LOS. Therefore it is very hard to achieve high performance with a single filter in both LOS and NLOS environments. In this paper, a localization method using a robust extended Kalman filter and track-quality-based (REKF-TQ) fusion algorithm is proposed to mitigate the effect of NLOS errors. Firstly, the EKF and REKF are used in parallel to obtain the location estimates of mobile nodes. After that, we regard the position estimates as observation vectors, which can be implemented to calculate the residuals in the Kalman filter (KF) process. Then two KFs with a new observation vector and equation are used to further filter the estimates, respectively. At last, the acquired position estimates are combined by the fusion algorithm based on the track quality to get the final position vector of mobile node, which will serve as the state vector of both KFs at the next time step. Simulation results illustrate that the TQ-REKF algorithm yields better positioning accuracy than the EKF and REKF in the NLOS environment. Moreover, the proposed algorithm achieves higher accuracy than interacting multiple model algorithm (IMM) with EKF and REKF.

Fast non-intrusive estimation of liquid-phase interface and boundary heat flux in participating media by the Kalman filtering methods

State of Charge (SOC) estimation plays a crucial role in managing the performance and longevity of electric vehicle (EV) batteries. Various estimation techniques, including various Kalman Filters (KF), have been employed for this purpose. The mathematical process and limits of different KF family algorithms are analysed. In this study, an impact of nonlinear KF parameters on SOC estimation accuracy for EV batteries is analyzed. Specifically, the performance of Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF), and Particle Filter (PF) is examined. EKF is most preferred for SOC estimation whereas introducing sigma points in UKF is the basic difference between EKF and UKF. In comparison to EKF and UKF, PF is more flexible since it applies to nonlinear systems with non-Gaussian distributions. Utilising a sequence of noisy measurements, KF is a recursive method that approximates the state of a dynamic system. It requires accurate parameters such as process noise covariance (Q) and measurement noise covariance (R) to work effectively. In this work, the effect of these parameters on state of charge estimation using nonlinear KFs has been analysed under different settings of noise covariance matrices. Tuning these parameters is essential for optimizing the performance of the KF and ensuring accurate and reliable state estimation in real-world applications. To find parameter values that provide the best balance between accuracy, robustness, and computational efficiency, the tuning process involves experimentation, simulation, and validation using real-world data. According to the results observed, the filter parameters such as three diagonal elements of Q are selected as 2e-8, 0.005, 0.003 and R as 2e-6. By evaluating the state estimation accuracy under different driving scenarios, robustness and sensitivity of filtering technique in estimating SOC for EV batteries has been studied. This analysis contributes to the optimization of SOC estimation algorithms, enhancing the reliability and efficiency of EV battery management systems.

In the processing of radar data, target estimation states fusion is an important problem. Especially under the decentralized model, various estimations of target state which come from local filters with Kalman filter may be fused in order to obtain a more accurate target estimation state. There are two problems to be solved. The first is the fusion criterion, different fusion criteria bring out different results. Three fusion criteria are discussed; one is based on the covariance matrix of estimation error, the second is dependent on the infinite normal number, and the last is related to the element of the state vector. The second problem is the tracking algorithm; it affects the accuracy of the state fusion result. The three criteria and their corresponding algorithms are presented. The algorithms estimating target state are set up with a strong tracking filter other than Kalman or extended Kalman filter; they have better tracking performance so that they can overcome the deficiencies related to Kalman or extended Kalman filters. The reason for this is explained. Comparison results are given.

The Kalman filter is an optimal state estimator used in many of the engineering applications. In general, Kalman filter has two typical forms for non-linear estimation they are, the extended Kalman filter and unscented Kalman filter. This paper discusses the implementation of GPS-based Inertial navigation system (a combination of accelerometers and gyroscopes) using an extended Kalman filter (EKF). Both GPS and INS sensors are used to update the position and velocity in a land vehicle so that real time guidance and navigation tasks are performed. Due to the low sampling rate of GPS, continuous and reliable position cannot be obtained regularly. Similarly, the INS output tends to accumulate error over time which makes it unreliable. These limitations are overcome by integrating these systems due to their complementary characteristics in order to achieve long term accuracy. The method discussed involves collecting the data from GPS and INS sensors and processing them using EKF to obtain better estimates on the position and velocity. The vehicle orientation is linearized using the quaternion in the prediction and correction stages of EKF.

Traffic state estimation problems must deal with imperfect information. Imperfect information may occur as a result of misalignments between reality and the assumptions or simplifications made when developing the models as well as incomplete or corrupted data. One of the well-known methods of handling imperfect information is the Kalman filter algorithm and its variations, such as the extended Kalman filter and the unscented Kalman filter. While the standard Kalman filter and the extended Kalman filter have been widely applied to the state estimation of linear systems, the unscented Kalman filter has been reported the better choice in the multiclass traffic state estimation. In many applications of Kalman filters to traffic estimation problems, the model and measurement noise covariance matrices are normally estimated. When there is a mismatch between the true and the assumed noise distribution, however, the filter often suffers from performance degradation and even divergence in certain situations. To this end, this paper presents a more efficient and accurate algorithm embedded in the unscented Kalman filter to simultaneously estimate the traffic state and the model noise distribution statistics. The proposed method is facilitated through a simultaneous update of the model noise covariance matrix in the predicted covariance equations of the unscented Kalman filter algorithm. It is found through simulation that the proposed algorithm may improve the model performance over the standard algorithm.

A new estimation algorithm is derived and appraised for nonlinear systems. The notion and measures of information are defined and this leads to a discussion of the algebraic equivalent of the Kalman filter, the linear information filter. Examples of dynamic systems are simulated to illustrate the algebraic equivalence of the Kalman and information filters. The benefits of information space are also explored. Estimation for systems with nonlinearities is then considered starting with the extended Kalman filter. Linear information space is extended to nonlinear information space by deriving the extended information filter. The advantages of the extended information filter over the extended Kalman filter are demonstrated for systems involving both nonlinear state evolution and nonlinear observations.

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. These include estimating the state of a nonlinear dynamic system as well estimating parameters for nonlinear system identification (eg, learning the weights of a neural network). The EKF applies the standard linear Kalman filter methodology to a linearization of the true nonlinear system. This approach is sub-optimal, and can easily lead to divergence. Julier et al. (1997), proposed the unscented Kalman filter (UKF) as a derivative-free alternative to the extended Kalman filter in the framework of state estimation. This was extended to parameter estimation by Wan and Van der Merwe et al., (2000). The UKF consistently outperforms the EKF in terms of prediction and estimation error, at an equal computational complexity of (OL/sup 3/)/sup l/ for general state-space problems. When the EKF is applied to parameter estimation, the special form of the state-space equations allows for an O(L/sup 2/) implementation. This paper introduces the square-root unscented Kalman filter (SR-UKF) which is also O(L/sup 3/) for general state estimation and O(L/sup 2/) for parameter estimation (note the original formulation of the UKF for parameter-estimation was O(L/sup 3/)). In addition, the square-root forms have the added benefit of numerical stability and guaranteed positive semi-definiteness of the state covariances.

The unscented Kalman filter (UKF) has become a popular alternative to the extended Kalman filter (EKF) during the last decade. UKF propagates the so called sigma points by function evaluations using the unscented transformation (UT), and this is at first glance very different from the standard EKF algorithm which is based on a linearized model. The claimed advantages with UKF are that it propagates the first two moments of the posterior distribution and that it does not require gradients of the system model. We point out several less known links between EKF and UKF in terms of two conceptually different implementations of the Kalman filter: the standard one based on the discrete Riccati equation, and one based on a formula on conditional expectations that does not involve an explicit Riccati equation. First, it is shown that the sigma point function evaluations can be used in the classical EKF rather than an explicitly linearized model. Second, a less cited version of the EKF based on a second-order Taylor expansion is shown to be quite closely related to UKF. The different algorithms and results are illustrated with examples inspired by core observation models in target tracking and sensor network applications.