Nonlinear Kalman Filter Research Articles

Estimating the Purse Seine Net Geometry during a Hauling Operation Using a Data Assimilation Method

The dynamics of fishing nets can be estimated by modeling and numerically computing the forces acting on them. However, the dynamic models of fishing nets are highly nonlinear owing to the significant influence of hydrodynamic forces acting on the net. Therefore, if there are unknown parameters that define the state of motion in the model, it is often difficult to achieve high accuracy in the numerical simulations of fishing gear and evaluate its dynamics. To address this issue, a method is proposed for estimating these unknown parameters by integrating a nonlinear Kalman filter into a fishing net dynamics model. This study aimed to estimate the hauling velocity of large- and medium-sized purse seine fishing nets, which can be a challenging parameter to measure. The calculations are based on the data obtained from a research operation conducted by the Marine Fisheries Research and Development Center in 2019 using the purse seine fishing vessel “Taikei Maru No. 1”. The time series of the hauling-net velocity was estimated based on the results of the estimation experiment. These results allowed the estimation of the hauling velocity and calculation of the net dynamics during the hauling process. This shows that net dynamics simulation is possible even with unknown parameters.

A generic fusion framework integrating deep learning and Kalman filter for state of charge estimation of lithium-ion batteries: Analysis and comparison

Lithium-ion batteries (LIBs) are extensively utilized in power and energy storage systems. Accurately estimating state of charge (SOC) of LIBs using single methods is still challenging to fully address the problem associated with SOC. This paper proposes a novel generic fusion framework that integrates deep learning (DL) and Kalman filter (KF), characterized by an effective fusion of the results obtained by multiple-methods. Within the DL-based method, a spatial-temporal awareness model and an improved dung beetle optimizer are proposed to enhance the capture of spatial-temporal features and optimize hyperparameters. Additionally, three adaptive nonlinear KFs are employed for SOC estimation based on a simplified electrochemical model, with the adaptive cubature KF emerging as the most suitable following comparative analysis. Subsequently, the results from both approaches are integrated to derive the final result using various fusion methodologies. The effectiveness of the fusion framework is validated using experimental data, with kernel ridge regression emerging as the optimal fusion method. The best fusion results yield a mean absolute error of 0.2341% and a root mean square error of 0.3072%, demonstrating that proposed framework can effectively process and utilize multiple-source information, thereby enhancing accuracy and robustness compared to single methods and exhibiting adaptability to complex situations.

Developed square-root cubature Kalman filter-based solution for improving power system state estimation with unknown inputs and non-Gaussian noise

Understanding the ever-changing dynamics of power systems is crucial, and dynamic state estimation (DSE) plays a vital role in achieving this. However, traditional nonlinear Kalman filters (NKFs) face limitations: lack of access to control inputs and presence of non-Gaussian noise in measurements, impacting their accuracy and robustness. This research introduces a novel robust DSE method that tackles these challenges head-on. For the first time in DSE, it leverages the predictive power of Holt-Winters Triple Exponential Smoothing to model the time-varying behavior of control inputs. This innovative approach allows for the simultaneous estimation of dynamic state variables such as the rotor angle and rotor speed changes, as well as transient voltages and control inputs like mechanical input torque and excitation voltage, even in the presence of non-Gaussian noise. Furthermore, the method employs modified projection statistics and a Cauchy function. This unique combination effectively bounds the influence of observation outliers while maintaining high statistical estimation efficiency. This innovative approach utilizes a square cubature Kalman filter (SCKF) for enhanced numerical stability. Extensive simulations under various anomalous conditions demonstrate the method's superior accuracy and efficiency in estimating the state vector. These results highlight its potential to significantly improve power system estimation and pave the way for real-time applications.

Joint Bayesian estimation of process and measurement noise statistics in nonlinear Kalman filtering

Bayesian filtering aims at sequentially estimating the states and parameters in nonlinear dynamical systems, and finds applications in a variety of engineering fields. Extended and unscented Kalman filter are popular choices for this purpose as they typically achieve a good trade-off between computational complexity and accuracy, while still quantifying posterior uncertainty. However, the accuracy of the identified states and parameters, both in terms of their posterior mean and variance, is heavily influenced by the assumed statistics of both the process and measurement noise terms. Therefore, learning the noise characteristics becomes an essential aspect of Bayesian filtering, but is non-trivial especially in cases where process and measurement noise statistics are jointly non-identifiable. This article proposes a methodology that decomposes measurement and process noise learning in a fully Bayesian setting. For measurement noise estimation, we improve upon a Bayesian method that leverages conjugacy of the inverse-Wishart distribution with respect to the Gaussian likelihood model. For process noise estimation, Bayesian model averaging is coupled with a mutation scheme to compare the performance of competing models while exploring high-probability regions in the space of the unknown process noise variance. The algorithm is first tested on simple numerical models for both state estimation and parameter learning, highlighting the superior accuracy of the proposed measurement noise correction and the benefits of decoupling process and measurement noise learning in non-identifiable settings. Then we demonstrate the usefulness of this algorithm in more challenging problems, namely identification of a multi-degree-of-freedom system under unmeasured excitation, and modeling of the highly nonlinear response of a full-scale bridge pier using experimental data.

Nonlinear Kalman filter for gyroscopic and accelerometer noise rejection of an unmanned aerial vehicle control strategy

This study addresses timing issues inherent in traditional proportional-integral-derivative (PID) controllers for drone angle control and introduces an innovative solution, the adaptive PID flight controller, aimed at optimizing PID gains for improved performance in terms of speed, accuracy, and stability. To enhance the controller's robustness against noise and accurately estimate the system's state, a Kalman filter is incorporated. This filtering mechanism is designed to reject noise and provide precise state estimation, thereby contributing to the overall effectiveness of the adaptive PID flight controller in managing altitude dynamics for unmanned aerial vehicles (UAVs). The comparative methodology evaluates three configurations: a single PID controller for all three angles, two PID controllers dedicated to pitch/roll and yaw angles separately, and three PID sub-controllers for each angle (pitch, roll, and yaw). The study seeks to identify the most effective PID configuration in terms of stability, responsiveness, and accuracy while highlighting the added benefits of noise rejection and state estimation through the Kalman filter. This integrated approach showcases innovation and effectiveness, introducing a comprehensive solution not explored in previous research.

Adaptive nonlinear Kalman filters based on credibility theory with noise correlation

To solve the divergence problem and overcome the difficulty in guaranteeing filtering accuracy during estimation of the process noise covariance or the measurement noise covariance with traditional new information-based nonlinear filtering methods, we design a new method for estimating noise statistical characteristics of nonlinear systems based on the credibility Kalman Filter (KF) theory considering noise correlation. This method first extends credibility to the Unscented Kalman Filter (UKF) and Extended Kalman Filter (EKF) based on the credibility theory. Further, an optimization model for nonlinear credibility under noise related conditions is established considering noise correlation. A combination of filtering smoothing and credibility iteration formula is used to improve the real-time performance of the nonlinear adaptive credibility KF algorithm, further expanding its application scenarios, and the derivation process of the formula theory is provided. Finally, the performance of the nonlinear credibility filtering algorithm is simulated and analyzed from multiple perspectives, and a comparative analysis conducted on specific experimental data. The simulation and experimental results show that the proposed credibility EKF and credibility UKF algorithms can estimate the noise covariance more accurately and effectively with lower average estimation time than traditional methods, indicating that the proposed algorithm has stable estimation performance and good real-time performance.

Online state estimation of Li-ion batteries using continuous-discrete nonlinear Kalman filters based on a nonlinear simplified electrochemical model

Online monitoring of Lithium-ion batteries (LIBs) during the charge and discharge periods is imperative to characterize their performance, assess their safety, and mitigate their degradation. One way to achieve this goal is to resort to model-based state estimations. Electrochemical based modelling has demonstrated significant potential to predict both the microscopic and the macroscopic battery dynamics. Existing electrochemical models are however continuous-time systems while measurements are performed at discrete time steps. Yet, most estimators are designed based on a fully discretized model. In this study, various types of continuous-discrete nonlinear Kalman filters are developed not only to address this challenge but also to compare their performance in utilizing highly nonlinear and meaningful physical variables. The filters rest on a simplified electrochemical model that accounts for the nonlinearities that stem from the dependence of the parameters on the electrolyte concentration. The performance of the estimators in adaptive and nonadaptive modes is tested for two different driving schedule loads (FUDS and Synthetic random higher C-rate). The results reveal that the estimators can track the real states' values if the filter model matches the real battery, even in the presence of 30 % error in the states' initial conditions. Furthermore, unlike the continuous-discrete extended Kalman filter (CD-EKF), the suggested continuous-discrete unscented Kalman filter (CD-UKF) effectively predicts the actual state values, even in instances where a significant structural mismatch exists between the embedded model and the synthetically generated measurement signal.

Forecasting hourly WTI oil front monthly price volatility densities

<p>Financial commodity markets have an impact on company values and cash flows, where price movements within frequent time intervals can be both significant and random. Understanding highly frequent price movements is both important and difficult. In this paper, I measured and forecasted volatility for high-frequency (mostly twelve hours per day) WTI Oil front month price movements from 2012 to 2024 (about 40,500 observations). I created a new stochastic volatility model for extracting latent volatility using the non-linear Kalman filter. Stable and strictly ergodic hourly price series, along with the BIC optimal non-linear (generic) general method of moments model coefficients, enabled this process. The latent volatility seemed to separate into two volatility factors. One factor that was very persistent suggested slow mean reversion, and one choppy strongly mean-reverting factor. The data dependence found in the factor volatility series suggested forecasting ability. I applied classical static forecasts and three machine learning regression techniques to predict one-step-ahead volatility. The two volatility factors and one summarizing exponential factor were reported, along with several fit measures, including the root mean square error and Theil's measure of covariance.</p>

Study of the influence of the parametric identifiability degree on the estimation accuracy of navigation system errors

The high-precision correction problem of navigation information for a small unmanned aerial vehicle flying with altitude changes over a wide range is studied. Numerical criteria of parametric identifiability degreefor one class of nonlinear systems represented by the SDC method are considered. Error models for inertial navigation system of various detail levels are studied and the results of mathematical and semi-natural modeling of a nonlinear Kalman filter with different models are presented. Keywords small unmanned aerial vehicle, inertial navigation system, errors, correction, degree of parametric identifiability, nonlinear Kalman filter, SDC method

Iterated Orthogonal Simplex Cubature Kalman Filter and Its Applications in Target Tracking

In order to increase a nonlinear system’s state estimate precision, an iterated orthogonal simplex cubature Kalman filter (IOSCKF) is presented in this study for target tracking. The Gaussian-weighted integral is decomposed into a spherical integral and a radial integral, which are approximated using the spherical simplex-radial rule and second-order Gauss–Laguerre quadrature rule, respectively, and result in the novel simplex cubature rule. To decrease the high-order error terms, cubature points with appropriate weights are taken from the cubature rule and processed using the provided orthogonal matrix. The structure supporting the nonlinear Kalman filter incorporates the altered points and weights and the calculation steps; from this, the updated time and measurement can be inferred. The Gauss–Newton iteration is employed repeatedly to adjust the measurement update until the termination condition is met and the IOSCKF is attained. The proposed algorithms are applied in target tracking, including CV target tracking and spacecraft orbit tracking, and the simulation results reveal that the IOSCKF can achieve higher accuracy compared to the CKF, SCKF, and OSCKF. In spacecraft orbit tracking simulation, compared with the SCKF, the position tracking accuracy and velocity tracking accuracy of the OSCKF are increased by 2.21% and 1.94%, respectively, which indicates that the orthogonal transformation can improve the tracking accuracy. Furthermore, compared with the OSCKF, the position tracking accuracy and velocity tracking accuracy of the IOSCKF are increased by 2.71% and 2.97%, respectively, which indicates that the tracking accuracy can be effectively improved by introducing iterative calculation into the measurement equation, thus verifying the effectiveness of the method presented in this paper.

A study on the implementation of nonlinear Kalman filter applying MMG model

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.

Development and Dynamic State Estimation for Robotic Knee–Ankle Orthosis With Shape Memory Alloy Actuators

Abstract The development of rehabilitation robots has long been an issue of increasing interest in a wide range of fields. An important aspect of the ongoing research field is applying flexible components to rehabilitation equipment to enhance human−machine interaction. Another major challenge is to accurately estimate the individual’s intention to achieve safe operation and efficient training. In this article, a robotic knee−ankle orthosis (KAO) with shape memory alloy (SMA) actuators is developed, and the estimation method is proposed to determine the joint torque. First, based on the analysis of human lower limb structure and walking patterns, the mechanical design of the KAO that can achieve various rehabilitation training modes is detailed. Next, the dynamic model of the hybrid-driven KAO is established using the thermodynamic constitutive equation and Lagrange formalism. In addition, the joint torque estimation is realized by the nonlinear Kalman filter method. Finally, the prototype and human subject experiments are conducted, and the experimental results demonstrate that the KAO can assist lower limb movements. In the three experimental scenarios, reductions of 59.1%, 16.5%, and 73% of the torque estimation error during the knee joint movement are observed, respectively.

Brain model state space reconstruction using an LSTM neural network

Objective. Kalman filtering has previously been applied to track neural model states and parameters, particularly at the scale relevant to electroencephalography (EEG). However, this approach lacks a reliable method to determine the initial filter conditions and assumes that the distribution of states remains Gaussian. This study presents an alternative, data-driven method to track the states and parameters of neural mass models (NMMs) from EEG recordings using deep learning techniques, specifically a long short-term memory (LSTM) neural network. Approach. An LSTM filter was trained on simulated EEG data generated by a NMM using a wide range of parameters. With an appropriately customised loss function, the LSTM filter can learn the behaviour of NMMs. As a result, it can output the state vector and parameters of NMMs given observation data as the input. Main results. Test results using simulated data yielded correlations with R squared of around 0.99 and verified that the method is robust to noise and can be more accurate than a nonlinear Kalman filter when the initial conditions of the Kalman filter are not accurate. As an example of real-world application, the LSTM filter was also applied to real EEG data that included epileptic seizures, and revealed changes in connectivity strength parameters at the beginnings of seizures. Significance. Tracking the state vector and parameters of mathematical brain models is of great importance in the area of brain modelling, monitoring, imaging and control. This approach has no need to specify the initial state vector and parameters, which is very difficult to do in practice because many of the variables being estimated cannot be measured directly in physiological experiments. This method may be applied using any NMM and, therefore, provides a general, novel, efficient approach to estimate brain model variables that are often difficult to measure.

Flatness-based disturbance observer for condition monitoring of marine power generation units

The article proposes a flatness-based disturbance observer for condition monitoring of the integrated power generation system, which comprises a marine turbine and a synchronous generator. It is proven that this power generation system is differentially flat, thus it can be transformed into an input–output linearized form and into the canonical Brunovsky form. For the linearized state-space description of the system, state estimation can be performed with the use of Kalman Filtering (derivative-free nonlinear Kalman filter). Moreover, by re-designing the Kalman filter as a disturbance observer, it becomes possible to estimate disturbances, model uncertainty terms and fault-induced perturbations that affect the power unit. Consequently, model inconsistencies and faulty conditions in the functioning of the marine power unit can be diagnosed.

A novel adaptive nonlinear Kalman filter scheme for DVL-aided SINS alignment in underwater vehicles

The rapid and accurate initial alignment of a strapdown inertial navigation system (SINS) aided by a Doppler velocity logger (DVL) still poses a significant challenge in unmanned underwater vehicle navigation. In this study, a novel adaptive nonlinear filtering alignment scheme is developed for a SINS with a large misalignment angle. In previous studies, the frame-inconsistency problem of the nonlinear SINS error model was neglected. Thus, a modified nonlinear error model of a SINS is derived here rigorously as the process model of the nonlinear Kalman filter. In addition, the noise amplification of applying Sage-Husa adaptive Kalman filter (SHAKF) in alignment is analyzed, and a novel arctangent fading memorial factor is introduced to dynamically adjust the weight of current measurements. Simulations and field experiments indicate that the proposed algorithm is efficient in underwater DVL/SINS in-motion alignment. The alignment accuracy and convergence rate of the modified model are significantly improved compared to those of the traditional model. The arctangent fading memorial SHAKF can reliably estimate the measurement noise and improve the performance of alignment.

Investigation into the nonlinear Kalman filter to correct the INS/GNSS integrated navigation system

Investigation into the nonlinear Kalman filter to correct the INS/GNSS integrated navigation system

Adaptive unscented Kalman filter for neuronal state and parameter estimation.

Data assimilation techniques for state and parameter estimation are frequently applied in the context of computational neuroscience. In this work, we show how an adaptive variant of the unscented Kalman filter (UKF) performs on the tracking of a conductance-based neuron model. Unlike standard recursive filter implementations, the robust adaptive unscented Kalman filter (RAUKF) jointly estimates the states and parameters of the neuronal model while adjusting noise covariance matrices online based on innovation and residual information. We benchmark the adaptive filter's performance against existing nonlinear Kalman filters and explore the sensitivity of the filter parameters to the system being modelled. To evaluate the robustness of the proposed solution, we simulate practical settings that challenge tracking performance, such as a model mismatch and measurement faults. Compared to standard variants of the Kalman filter the adaptive variant implemented here is more accurate and robust to faults.

Trajectory Planning for Multiple UAVs and Hierarchical Collision Avoidance Based on Nonlinear Kalman Filters

Fully autonomous trajectory planning for multiple unmanned aerial vehicles (UAVs) is significant for building the next generation of the logistics industry without human control. This paper presents a method to enable multiple UAVs to fly in the same trajectory without collision. It benefits several applications, such as smart cities and transfer goods, during the COVID-19 pandemic. Different types of nonlinear state estimation are deployed to test the position estimation of drones by treating the information from AirSim as offline dynamic data. The obtained global positioning system sensor data and magnetometer sensor data are determined as the measurement model. The experiment in the simulation is separated into (1) the localization state, (2) the rendezvous state, in which the proposed rendezvous strategy is presented by using the relation between velocity and displacement through the setting area, and (3) the full mission state, which combines both the localization and rendezvous states. The localization state results show the best RMSE in the case of full GPS available at 0.21477 m and 0.25842 m in the case of a GPS outage during a period of time by implementing the ensemble Kalman filter. Similarly, the ensemble Kalman filter performs well with an RMSE of 0.5112414 m in the rendezvous state and demonstrates exceptional performance in the full mission state. Moreover, the experiment is implemented in a real-world situation with some basic drone kits as proof that the proposed rendezvous strategy can truly operate.

An experimental comparative study of adaptive sigma-point Kalman filters: Case study of a rigid–flexible four-bar linkage mechanism and a servo-hydraulic actuator

This paper investigates the application, design, and implementation of different adaptive sigma-point Kalman filters for nonlinear dynamic systems. It is a well-known fact that the performance of nonlinear Kalman filters depends on a prior knowledge of system noise statistics. In real-time applications, this knowledge is difficult and in some cases impossible to predefine. Therefore, an adaptation mechanism based on the random weighting method is proposed in this work to estimate, simultaneously, noise covariance and the state of the system. On the other hand, for real time execution, the computational efficiency of such filters should also be considered to minimize the burden on computational resources. Based on these two factors, in this paper, four different variants of the adaptive sigma-point Kalman filter are studied in detail: (I) an Adaptive Unscented Kalman filter (AUKF), (II) an Adaptive Square-root Unscented Kalman filter (ASr-UKF) with an Unscented Transformation sigma-point selection strategy, (III) an Adaptive Scaled Spherical Simplex Unscented Kalman filter (AS3UKF), and (IV) an Adaptive Square-root Scaled Spherical Simplex Unscented Kalman filter (ASr-S3UKF) with scaled spherical simplex sigma-point selection strategy. The performance of these adaptive sigma-point Kalman filters is analyzed here in depth and discussed for application to a simulated rigid–flexible four-bar linkage mechanism and a practical servo-hydraulic actuator setup under different scenarios. The simulation and the experimental results indicated that, for the cases studied in this article, the filters are competitive and present advantages and disadvantages that should be dealt with according to the requirements of the problem.

State estimation accuracy analysis of an induction electric drive by the algorithms of Luenberger and Kalman

In complex electromechanical objects containing electric drives with induction motors, it is often difficult or impossible to install sensors of output variables. In this case, to determine the output coordinates of the motor, it is necessary to introduce state observers into the control system of the electric drive. The main problems of creating observers are the presence of noise and interference in the measuring circuits of the control system which affect the accuracy of the estimation of immeasurable state variables. The paper presents a comparison of the accuracy of estimates obtained as a result of the work of the observer algorithms based on the Kalman filter and the observer of the Luenberger in the induction electric drive system, with vector control at the noise level of the current measurement channels in the stator windings of the induction motor. To synthesize algorithms for state observers, methods of identification theory and quasi-linearization of nonlinear models of the control object under consideration were used. The simulation model of an induction motor is based on a classical vector field-oriented control system where an estimate of the angular speed of the motor shaft is used as a feedback signal. The model implements the following blocks: a mathematical model of an induction motor in a two-phase fixed coordinate system α–β; the structure of the observer algorithm; the procedure for converting the basis of the current vector and the control voltage from stationary to rotating and vice versa; proportional-integral regulators of current, flux linkage and angular speed. The S-shaped intensity setter forms a speed setting curve. The input signals for observers are the stator voltages and currents of the reference model of an induction motor. The adaptation coefficients for the Luenberger observer were selected experimentally from the condition of obtaining the average minimum value of the difference modulus of the estimated values. The covariance matrices for the observer based on the Kalman filter are configured on the basis of the experiment, ensuring a minimum of the average value of the absolute error. The time dependences of the transients of the angular speed of the shaft, the modulus of the flux linkage vectors of the rotor and stator currents are obtained. The dependencies were evaluated when starting an induction motor with nominal values and values of frequency and voltage amounting to 10 % of the nominal values. The work of estimation algorithms in the presence of a noise component, as well as when changing the parameters of the induction motor replacement circuit by ± 10 %, is investigated. The results of modeling the operation of the electric drive in starting modes with a mechanical load equal to the nominal value at a supply voltage frequency of 50 Hz and at 10 % of the nominal value for a voltage of 1 Hz are obtained. It is shown that the greatest relative estimation errors occur in the starting mode of the electric drive, and the maximum accuracy is achieved in the case of using a nonlinear Kalman filter. The results of the work can be used in the development of automatic control systems for sensorless electric drives and frequency-controlled electric drive of centrifugal pumping units for oil production.