On computational robustness of accurate continuous-discrete unscented Kalman filtering for target tracking models

MATLAB-based general approach for square-root extended-unscented and fifth-degree cubature Kalman filtering methods

This article examines an economic growth model that expresses the interaction between production, technology stock, and research and development (R&D) investments. The goal of this study is to maximize production. Considering the presence of Gaussian white noises, this model is reformulated as a stochastic optimal control problem, where the R&D investment rate is defined as the control input. We aim to explore the efficiency of Kalman filtering approaches for solving this problem. Here, the extended Kalman filter (EKF) and unscented Kalman filter (UKF) are applied for state estimation. The state equation linearization is made in the EKF, while the unscented transform is taken in the UKF for generating a set of sigma points. These approaches aim to estimate the state dynamics from different perspectives. With these state estimates, two different computational algorithms are proposed, the EKF for state-control (EKF4SC) and UKF for state-control (UKF4SC) algorithms. The optimal control policy is designed to minimize the cost function. For illustration, the model's parameters are considered in the simulation experiment. The simulation results showed that the UKF has higher accuracy in the state estimation with the smallest mean squares of error compared with the EKF. Moreover, the optimal control policy based on the state estimate generated from the UKF could optimize the cost function of the problem. Hence, the results show the efficiency of the algorithms proposed. In conclusion, the application of the Kalman filtering algorithms to the economic growth model is presented. The significance of this study is to provide a stochastic optimal control model for the economic growth problem and to suggest an efficient computational approach for solving this stochastic optimal control problem.

A required assumption of a Kalman filter, the most-widely-used state estimator in avionic systems, is the Gaussian and whiteness of process and measurement noise. If the assumption fails, the performance of the Kalman filter degrades, and its estimation results are no longer optimal. In fact, many avionic applications produce colored noise, and the parameters of colored noise models are typically unknown beforehand without additional information about the noise statistics. In addition, the functions of each underlying model-nonlinear dynamic and measurement models - are sometimes improper or partially unknown. To estimate the states of systems with unknown correlations of each instance of noise and uncertain modeling errors of parametric models, we propose a novel approach that incorporates the kernel embedding of distributions into the extended Kalman filter. In our approach, kernel embedding maps process and measurement residuals, defined by differences between outputs of approximate system models and collected training data, into a reproducing kernel Hilbert space to generate nonparametric models in the functional space. Results from Monte Carlo simulations demonstrate that the proposed method, compared to existing methods (e.g., extended Kalman filter and Gaussian process-based filter), improves the accuracy of state estimation under colored noise conditions.

Path-following control systems for ships can be designed using both heading and course angle autopilots in conjunction with a proportional line-of-sight (LOS) guidance law. Ships are usually equipped with a gyrocompass from a safety perspective since magnetic compasses are susceptible to magnetic disturbances. Unfortunately, the gyrocompass is an expensive device, and smaller vessels and boats cannot afford to use this as the primary device for steering. An alternative solution is to compute the course over ground (COG) and speed over ground (SOG) from global navigation satellite systems (GNSS) and use these signals for feedback control. This article presents course autopilot design for path following and a five-state extended Kalman filter (EKF) to estimate the COG and SOG efficiently. Even though many algorithms exist for computation of the COG and SOG, it is advantageous to design an EKF since a state estimator can be extended to include other sensory systems such as Doppler Velocity Log (DVL), radar, attitude rate sensors, computer vision systems, etc. This is in contrast to proprietary systems that do not allow the user to modify the software. The convergence and accuracy of the EKF are significantly improved by using target-tracking models in combination with kinematic equations. A high-fidelity model of a MARINER class cargo ship is used in the path-following case study. From the simulation study, it can be concluded that the EKF successfully estimates the COG and the SOG from GNSS measurements during path following. The solution is remarkably robust and accurate, and when combined with a course autopilot, the need for a compass is eliminated during path following.

Adaptive interacting multiple model for underwater maneuvering target tracking with one-step randomly delayed measurements

Accurate cubature and extended Kalman filtering methods for estimating continuous-time nonlinear stochastic systems with discrete measurements

Data fusion algorithms are inevitable in improving the accuracy and reliability of radar target tracking. Considering the shortcomings of target tracking in Cartesian coordinates, all the fusion process are carried out in spherical coordinate system. To fuse data from bidimensional radar and Infrared Search and Track (IRST), five data fusion methods, Bayesian fusion, Signal variance based fusion, Optimal fusion, Error variance based fusion and Extended Kalman Filter (EKF) based fusion are proposed and analyzed. The performances of the methods are analyzed in terms of Root Mean Square Error in Position (RMSPE) and Mean Absolute Error in Position (MAPE) for a wide range of measurement noise variances. It is concluded that EKF based fusion has the best error performance. Kalman Filter State-Vector Fusion (KF-SVF) which is a hierarchical fusion architecture, having first level of fusion with Kalman filter and second level with error variance based fusion method, is available in the literature. In this work it has been expanded to Extended Kalman Filter—State-Vector Fusion (EKF-SVF) with either error variance based fusion, signal variance based fusion or optimal information fusion as second level. All the three EKF-SVF methods have similar error characteristics, but EKF-SVF with signal variance based fusion, unlike the other two, provides less computational cost since it does not involve matrix inversion. While comparing EKF based fusion and EKF-SVF, EKF-SVF offers increased error minimization.

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.

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.

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.

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 electrification of the transport sector increases the demand for a battery management system that allows the Li-ion battery to work in a sophisticated way to reach its maximum. The main focus of the battery management system is the estimation of the battery’s state of charge (SOC) which is an indicator to determine the driving range of an electric vehicle. However, battery modeling has a direct influence on SOC estimation. In this article, the equivalent circuit model (ECM) and extended Kalman filter (EKF) algorithm are combined for SOC estimation of the Li-ion battery. Since the performance of Kalman filter algorithms strictly depends on noise covariance matrices, incorrect values reduce the accuracy and make divergence in state estimation. Therefore, it is motivated to analyze the impact of process noise and measurement noise covariance matrix values on SOC estimation using Kalman filter algorithms. Ten different combinations of noise covariance matrices are employed to analyze the effect of these matrices on SOC estimation. As a result, the tuning of the filter is critical for improvement in state estimation regardless of the type of the battery model and filtering technique. The matrix values are chosen to take into account the convergence rate and estimation error. To validate the effectiveness of the chosen noise covariance matrices, the simulation results of the ECM based EKF estimation method is superimposed with the discharge curves available in the data sheet. The simulation results show minimal estimation error and a good convergence rate for tuned EKF.

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.

The extended Kalman filter (EKF)used in radar tracking applications is computationally intensive leading to difficulties in broadband real-time applications. Hence, it is necessary to resort to parallel processing techniques. With the advent of very large scale integration (VLSI) technology, different types of architectures such as systolic arrays and wavefront arrays have been developed for this purpose. The transputer is one such architecture that can be used as a processing node in a parallel processing network. Therefore, two configurations, namely, pipe configuration and mesh configuration are developed for transputer implementation of radar tracking extended Kalman filtering. The EKF algorithm is analyzed in detail for reduction of number of computations. The transputer implementations are written using ANSI-C language with the proposed reduction of matrix operations and run using a mother board with 4 TRAMS each containing one T800 transputer. A typical maneuvering trajectory is simulated and used to compare the computation time per iteration of single and four transputer implementations of the tracking Kalman filter. Further, the speed-up comparison of sequential implementation of EKF, both with and without the proposed reduction of computations, is reported.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Object-Oriented Environment for Assessing Tracking Algorithms