Iterated cubature Kalman filter and its application

The cubature Kalman filter (CKF) is well-known for a decade as a derivative-free nonlinear Kalman filter that is well-suited for high-dimensional nonlinear estimation problems. This paper further develops this classical CKF in order to cope with time-varying noise statistics as well as inequality constraints on the estimated states. The resulting adaptive filter is suggested to provide more accurate state estimates and to be more robust against filter divergence. Moreover, this contribution proposes an automated filter design based on numerical optimization which uses the normalized estimation error squared (NEES) and the normalized innovation squared (NIS) as part of the objective function. The novel adaptive CKF is applied to wind turbines in order to assess the potential improvement for state and parameter estimation. The simulation results for an illustrative acid test scenario with time-varying measurement noise show the superiority of the novel adaptive CKF since it compensates the noise robustly and thereby outperforms the classical filter.

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.

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.

Many technologies need to be established to realize autonomous ships. In particular, accurate state estimation in real time is one of the most important technologies. In the ship and ocean engineering fields, there have been many studies on state estimation using nonlinear Kalman filters. Several methods have been proposed for nonlinear Kalman filters. However, there is insufficient verification on the selection of which filter should be applied among them. Therefore, this study aims to validate the filter selection to provide a guideline for filter selection. The effects of modeling error, observation noise, and type of maneuvers on the estimation accuracy of the unscented Kalman filter (UKF) and ensemble Kalman filter (EnKF) used in this study were investigated. In addition, it was verified whether filtering could be performed in real time. The results show that modeling error significantly impacts the estimation accuracy of the UKF and EnKF. However, the observation noise and types of maneuvers did not have an impact like the modeling error. Thus, we obtained the guideline that UKF and EnKF should be used differently depending on the required computation time. We also obtained that keeping the modeling error sufficiently small is essential to improving the estimation accuracy.

Simultaneous state and parameter estimation is essential for control system design and dynamic modeling of physical systems. This capability provides critical real-time insight into system behavior, supports the discovery of underlying mechanisms, and facilitates adaptive control strategies. Surveyed in this review paper are two classes of state and parameter estimation methods: Kalman Filters and Luenberger Observers. The Kalman Filter framework, including its major variants such as the Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF), Cubature Kalman Filter (CKF), and Ensemble Kalman Filter (EnKF), has been widely applied for joint and dual estimation in linear and nonlinear systems under uncertainty. In parallel, Luenberger observers, typically used in deterministic settings, offer alternative approaches through high-gain, sliding mode, and adaptive observer structures. This review focuses on the theoretical foundations, algorithmic developments, and application domains of these methods and provides a comparative analysis of their advantages, limitations, and practical relevance across diverse engineering scenarios.

At present, multiple unmanned aerial vehicles (UAVs) collaboration have attracted people’s more and more attention. The abundant relative distance information in UAV cooperative network can be employed to further improve UAV’s positioning accuracy. However, the traditional distance-based cooperative localization (CL) algorithm does not consider the performance of cooperative navigation information. Cooperative information with poor navigation performance will affect follower UAV’s state estimation accuracy. Thus, this paper proposes a weight factor graph CL method based on navigation performance evaluation. A novel navigation performance evaluation strategy based on Fisher information and relative entropy is proposed. The distance observations and cooperative nodes’ positions with higher estimation accuracy and contribution are selected to estimate follower UAV’s position. Considering the difference contributions of the selected cooperative information to follower UAV’s state estimation, we employ the information weight processing to achieve higher state estimation accuracy. Based on the selected cooperative information and weight coefficient, a multi-UAVs cooperative optimization graph model based on navigation performance is constructed. The global optimization cost function is constructed using relative distance constraints from the current time to historical time in the sliding window. The global optimal navigation solution of follower UAV is achieved using iterative weighted least squares method. Simulation and experiment results show that the proposed navigation performance evaluation strategy can effectively screen cooperative information with higher estimation accuracy and contribution for CL. Using the selected high-quality cooperative information, the system computational complexity is reduced without significantly affecting system positioning accuracy.

Flying vehicle’s navigation, direction, and control in real‐time results in the design of a strap‐down inertial navigation system (INS). The strategy results in low accuracy, performance with correctness. Aiming at the attitude estimation problem, many data fusion or filtering methods had been applied, which fail in many cases, which attains the nonlinear measurement model, process dynamics, and high navigation range. The main problem in unmanned aerial vehicles (UAVs) and flying vehicles is the determination of attitude angles. A novel attitude estimation algorithm is proposed in this study for the unmanned aerial vehicle (UAV). This research article designs two filtering algorithms for fixed‐wing UAVs which are nonlinear for the attitude estimation. The filters are based on Kalman filters. The unscented Kalman filter (UKF) and cubature Kalman filter (CKF) were designed with different parameterizations of attitude, i.e., Euler angle (EA) and INS/unit quaternion (UQ) simultaneously. These filters, EA‐UKF and INS‐CKF, use the nonlinear process and measurement model. The computational results show that among both filters, the CKF attains a high accuracy, robustness, and estimation for the attitude estimation of the fixed‐wing UAV.

This paper compares state estimation techniques for nonlinear stochastic dynamic systems, which are important for target tracking. Recently, several methods for nonlinear state estimation have appeared utilizing various random-point-based approximations for global filters (e.g., particle filter and ensemble Kalman filter) and local filters (e.g., Monte-Carlo Kalman filter and stochastic integration filters). A special emphasis is placed on derivations, algorithms, and commonalities of these filters. All filters described are put into a common framework, and it is proved that within a single iteration, they provide asymptotically equivalent results. Additionally, some deterministic-point-based filters (e.g., unscented Kalman filter, cubature Kalman filter, and quadrature Kalman filter) are shown to be special cases of a random-point-based filter. The paper demonstrates and compares the filters in three examples, a random variable transformation, re-entry vehicle tracking, and bearings-only tracking. The results show that the stochastic integration filter provides better accuracy than the Monte-Carlo Kalman filter and the ensemble Kalman filter with lower computational costs.

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.

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.

The Kalman-filter-based algorithms as the mainstream algorithms of dynamic state estimation of power systems have been extensively used to provide accurate data for power system applications. However, few comparisons are made to show their advantages and disadvantages. In this paper, four Kalman-filter-based algorithms (i.e., extended Kalman filter, unscented Kalman filter, cubature Kalman filter, and ensemble Kalman filter) are compared to show their differences from implementation complexity, estimation accuracy and calculation efficiency, the resistance to measurement errors, and the sensitivity to system scales. Finally, the simulation results on the 3-machine, 10-machine, and 48-machine power systems show their advantages and disadvantages.

Guest Editorial: Situational awareness of integrated energy systems

Estimation for state-of-charge of lithium-ion battery based on an adaptive high-degree cubature Kalman filter

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.

Accurate estimation of battery state of charge (SOC) is of great significance to improve battery management and service life. An unscented Kalman filter (UKF) method is used to increase the accuracy of SOC estimation in this paper. Firstly, a battery model that the parameters are identified by using the least squares algorithm is established, which is foundation of the two-order RC equivalent circuit model. Secondly, SOC is estimated by UKF. In order to validate the method, experiments have been carried out under different operating conditions for LiFePO4 batteries. The obtained results are compared with that of the extended Kalman filter. Finally, the comparison shows that the UKF method provides better accuracy in the battery SOC estimation. Its estimation error is less than 2%, which is better than EKF algorithm. An effective method is provided for state estimation for battery management system.