Nonlinear Kalman Filter Research Articles

Square Root Unscented Kalman Filter With Modified Measurement for Dynamic State Estimation of Power Systems

Dynamic state estimation of a power system provides essential information about its inherent dynamic change. Nonlinear Kalman filters (NKFs) have been identified as potential versatile tools in performing state estimations. One key challenge of using NKFs lies in the fact that the available observations are inaccurate due to the presence of non-Gaussian noise that degrades the filtering precision and robustness. Some robust NKFs are developed to correct the state estimation via using robust optimality criteria or approximating the posterior density. Considering that the essence in these robust NKFs is to weight the error covariance and noise variance, we propose a novel method that can directly act on measurement to weight error covariance and noise variance. The novel method is based on a square root unscented Kalman filter that can enhance the numerical stability. Then, a weighting factor is applied to the measurement model for alleviating the effect of abnormal measurement errors. The weighting factor is derived from consideration of an anisotropy covariance function and an upper envelope of the traditional “sinc” function for suppressing large measurement errors and retaining the information of very large errors. We call the new filter the square root unscented Kalman filter with modified measurement (SRUKF-MM). The proposed SRUKF-MM is applied to state estimation of the Western System Coordinating Council (WSCC) 3-machine system and the Northeastern Power Coordinating Council (NPCC) 48-machine system in a non-Gaussian noise environment. Simulation results show that the proposed method achieves significantly improved filtering performance compared to other related NKFs.

A Class of Nonlinear Kalman Filters Under a Generalized Measurement Model With False Data Injection Attacks

In this letter, a generalized measurement model (GMM) is established under the attack injecting additive and multiplicative false data, simultaneously. The GMM can be applied to any existing nonlinear Kalman filters (NKFs), such as extended Kalman filter (EKF), cubature Kalman filter (CKF), and geometric unscented Kalman filter (GUF). Based on the constructed GMM, a class of nonlinear Kalman filters with GMM (NKFs-GMM) is proposed to deal with additive false data, multiplicative false data, and simultaneous attacks on additive and multiplicative false data with no increase of computational complexity. Numerical simulations show that the filtering accuracy of NKFs-GMM is superior to that of corresponding NKFs.

Design and Implementation of Hybrid Adaptive Extended Kalman Filter for State Estimation of Induction Motor

Today, induction motor (IM) is still the most popular electrical machine due to its robust and rare element-free structure, lower maintenance requirement, and cost-effective production. State estimation for this motor is the cornerstone for speed-sensorless control, fault-tolerant control, and fault diagnostics. Nonlinear Kalman filters, especially extended Kalman filters (EKFs), are the most preferred state and/or parameter estimation methods for IM. However, they require a stochastic system with complete process and measurement noise covariances for optimal estimations. These noise covariances, unknown or partially known in practice, vary under different operating conditions of the IM. To deal with this problem, various adaptive EKFs (AEKFs) have been proposed, which can compensate for the effect of varying noise covariances, but each approach has its own pitfalls. This article discusses the hybrid AEKF (HAEKF), which eliminates the problems of existing AEKFs. To demonstrate its effectiveness, the proposed HAEKF is compared qualitatively and quantitatively with existing AEKFs through simulation and experimental studies. Finally, improved estimation stability and performance are provided with the proposed HAEKF observer.

Combined tracking system of GPS signals based on а nonlinear Kalman filter

Combined tracking system of GPS signals based on а nonlinear Kalman filter

WITHDRAWN: Nonlinear Kalman filters based rotor effective wind speed estimation: An experimental comparison

This article has been withdrawn at the request of the editor. The Publisher apologizes for any inconvenience this may cause. The full Elsevier Policy on Article Withdrawal can be found at http://www.elsevier.com/locate/withdrawalpolicy .

Systematic Estimation of Noise Statistics for Nonlinear Kalman Filters

We propose two new systematic and easy-to-implement online tuning strategies for nonlinear Kalman filters with low computational cost. The tuning strategies assume the process and measurement noise are due to parametric uncertainty. We assume nθ uncertain parameters which are translated into noise statistics by either i) generalized unscented transformation with 2nθ extra online model evaluations at every time step or ii) latin hypercube sampling, where the user sets the number of samples. Both approaches are distribution free, hence, the tuning strategies work for all kind of distributions. In the case study, it was found that the two proposed tuning strategies outperform the standard approach of fixed, diagonal noise matrices. In the case study, we further found that tuning based on the generalized unscented transformation seems to be more consistent than the method based on latin hypercube sampling for the same online computational cost. In addition, a Monte Carlo based tuning with modal noise adjustment is tested with promising performance. The modal noise adjustment is interesting as we can estimate the most likely point value of the noise (the mode of the noise distribution) and add this term to the state- and measurement equations at every time step.

The formation algorithm of errors models for satellite navigation systems and correction algorithm for aircraft navigation systems

An aircraft navigation complex as a part of an inertial navigation system, a satellite navigation system and an adaptive nonlinear Kalman filter is considered. In the Kalman filter, when the estimation process diverges, the model is constructed using an evolutionary algorithm. The used divergence indicators analyze the measured signals; the directly unmeasured state variables are not examined. A divergence criterion of directly unmeasured state variables is proposed on the grounds of the reduced measurements and the variances of the reduced measurement noises. The algorithm modification effectiveness is demonstrated by using the flight experiment data. Keywords navigation complex, adaptive nonlinear Kalman filter, divergence indicator, reduced measurements, flight experiment data

Adaptive Wide-Area Control of Power Systems Through Dynamic Load Modulating SVCs

This paper presents an adaptive wide-area control strategy based on energy function. The proposed method enhances the post-fault oscillation damping capability of Static Var Compensators (SVC) by incorporating load dynamics in the wide-area controller design that adapts to load variations. The load dynamics are obtained using Phasor Measurement Units (PMU) through load power and voltage data measurements, which are used to estimate a Moving Average (MA) dynamic load model. The obtained load model is then integrated into the wide-area controller to obtain the required SVC supplementary control action which modulates the load power in response to a disturbance based on inter-area dynamics obtained by employing a nonlinear Kalman filter and PMU measurements. The performance of the proposed strategy is evaluated on the simplified 14 generator model of the southeast Australian power system by using composite load models comprised of induction motor loads of various sizes and combinations. The obtained results show significant improvements in the system stability when the proposed control strategy is applied.

The Gaussian multiplicative approximation for state‐space models

Applications such as structural health monitoring (SHM) often rely on the analysis of time-series using methods such as state-space models (SSM). In this paper, we propose an analytical method called the Gaussian multiplicative approximation (GMA) that is applicable to multiplicative state-space models that are often encountered in practical SHM applications. The method enables the analytical inference of the mean vector and the covariance matrix for the product of two hidden states in the transition and/or observation models using linear estimation theory and the online estimation of model parameters as hidden states. The potential of combining the GMA and Bayesian dynamic linear models (BDLM) is illustrated through the development of (1) a generic component called online autoregressive that can estimate both the state variable and the parameter together; (2) a generic component called trend multiplicative for multiplicative seasonality model to identify non-harmonic periodic pattern whose amplitude changes linearly with time; and (3) a generic component called double kernel regression to identify non-harmonic periodic pattern that involves the product of two periodic kernel regression components. The SHM-based case studies presented confirm that the GMA exceeds the performance of the existing nonlinear Kalman filter methods in terms of accuracy along with the computational cost.

Simultaneous and sequential state and parameter estimation using receding-horizon nonlinear Kalman filter

Online estimation of internal states and parameters is often required for process monitoring, fault diagnosis and real time optimization. The conventional approach to tracking drifting parameters is to model their variations as a random walk process and estimate them simultaneously with the states. However, tuning of the random walk model is not a trivial exercise. Recently, moving window based simultaneous state and parameter estimation schemes have been proposed under the assumption that the parameters change slowly and remain constant within the window. These formulations make use of the conventional recursive Bayesian estimators or moving horizon estimator (MHE) for state estimation. While simultaneous smoothing of states along with filtering gives an edge to MHE over the conventional recursive estimators, MHE is computation intensive scheme. Recently, developed Receding-horizon Nonlinear Kalman (RNK) filter combines advantages of MHE (simultaneous smoothing) and recursive estimators (less computation time). In this work, we extend the RNK formulation to carry out simultaneous state and parameter estimation schemes under the assumption that parameters change at a slow rate and remain constant within the RNK window. Two maximum likelihood (ML) based parameter estimation schemes are developed by constructing partial likelihood and complete likelihood functions. Both formulations are further extended under the Bayesian framework to derive their maximum a-posteriori (MAP) versions by incorporating prior information of the parameters. The only tuning parameter in the proposed parameter estimation schemes is the window size, which is relatively easy to select. The efficacies of the proposed formulations are evaluated using simulation studies on two reactor systems and experimental data obtained from the benchmark quadruple tank system. Simulation studies reveal that the complete likelihood based estimation schemes have an edge over the partial likelihood based estimation schemes. Also, the proposed estimation schemes generate reasonably accurate state and parameter estimates with a significant reduction in the average computation time.

On SINS\\star tracker geo-localization

This work focuses on the feasibility of a fully autonomous geo-localization system for near-earth applications based on the strap-down inertial navigation system (SINS) and the star tracker. First, each sensor is analyzed individually. Then, the performance of the integrated system in a dynamic situation is investigated through extensive simulations. An in-depth analysis of the contributing error factors is given, and the impact on geo-localization estimation process is highlighted. The navigation solution is proven to be affected by the sensors’ errors plus an algorithmic error from the dead reckoning computation. Lastly, simulations are concluded to assess the dynamic movement scenario’s performance and navigational possibility using the nonlinear Kalman filter. The results show the continuing divergence of the integrated navigation system affected by the dead reckoning algorithm and the instability of the vertical channel. However, the continuous initial alignment in static mode reinitializes the position error successfully.

Advanced Controller Development Based on eFMI with Applications to Automotive Vertical Dynamics Control

High-level modeling languages facilitate system modeling and the development of control systems. This is mainly achieved by the automated handling of differential algebraic equations which describe the dynamics of the modeled systems across different physical domains. A wide selection of model libraries provides additional support to the modeling process. Nevertheless, deployment on embedded targets poses a challenge and usually requires manual modification and reimplementation of the control system. The novel proposed eFMI Standard (Functional Mock-up Interface for embedded systems) introduces a workflow and an automated toolchain to simplify the deployment of model-based control systems on embedded targets. This contribution describes the application and verification of the eFMI workflow using a vertical dynamics control problem with an automotive application as an example. The workflow is exemplified by a control system design process which is supported by the a-causal, multi-physical, high-level modeling language Modelica. In this process, the eFMI toolchain is applied to a model-based controller for semi-active dampers and demonstrated using an eFMI-based nonlinear prediction model within a nonlinear Kalman filter. The generated code was successfully tested in different validation steps on the dedicated embedded system. Additionally, tests with a low-volume production electronic control unit (ECU) in a series-produced car demonstrated the correct execution of the controller code under real-world conditions. The novelty of our approach is that it automatically derives an embedded software solution from a high-level multi-physical model with standardized eFMI methodology and tooling. We present one of the first full application scenarios (covering all aspects ranging from multi-physical modeling up to embedded target deployment) of the new eFMI tooling.

Reduced-Order Extended Kalman Filter for Deformable Tissue Simulation

Modelling of soft tissue deformation is a key issue in surgical simulation. Despite extensive research studies on this issue, accurate modelling of soft tissue deformation in run time still remains challenging. This paper proposes a new reduced-order nonlinear Kalman filter to emulate nonlinear behaviors of biological deformable tissues. This approach defines the deformable modelling problem as a reduced-order filtering problem to accurately calculate soft tissue deformation in real time. Soft tissue deformation is discretized in space using nonlinear finite element method based on hyperelasticity and further formulated as a nonlinear state-space equation for filtering estimation. Subsequently, the order of this nonlinear state-space equation is reduced using proper orthogonal decomposition to reduce the computational cost. Upon this reduced-order state-space equation, an extended Kalman filter is constructed to online calculate nonlinear behaviors of tissue physical deformation. Simulation results and comparison analysis prove the effectiveness of the suggested method for accurate simulation of tissue physical deformation in real time.

Forecasting Stochastic Volatility Characteristics for the Financial Fossil Oil Market Densities

This paper builds and implements multifactor stochastic volatility models for the international oil/energy markets (Brent oil and WTI oil) for the period 2011–2021. The main objective is to make step ahead volatility predictions for the front month contracts followed by an implication discussion for the market (differences) and observed data dependence important for market participants, implying predictability. The paper estimates multifactor stochastic volatility models for both contracts giving access to a long-simulated realization of the state vector with associated contract movements. The realization establishes a functional form of the conditional distributions, which are evaluated on observed data giving the conditional mean function for the volatility factors at the data points (nonlinear Kalman filter). For both Brent and WTI oil contracts, the first factor is a slow-moving persistent factor while the second factor is a fast-moving immediate mean reverting factor. The negative correlation between the mean and volatility suggests higher volatilities from negative price movements. The results indicate that holding volatility as an asset of its own is insurance against market crashes as well as being an excellent diversification instrument. Furthermore, the volatility data dependence is strong, indicating predictability. Hence, using the Kalman filter from a realization of an optimal multifactor SV model visualizes the latent step ahead volatility paths, and the data dependence gives access to accurate static forecasts. The results extend market transparency and make it easier to implement risk management including derivative trading (including swaps).

Detecting Epileptic Seizures Based on Parameter Estimation of a Neural Mass Model

This paper presents two adaptive Kalman filters (KFs) for nonlinear model updating where, in addition to nonlinear model parameters, the covariance matrix of measurement noise is estimated recursively in a near online manner. Two adaptive KF approaches are formulated based on the forgetting factor and the moving window covariance-matching techniques using residuals. Although the proposed adaptive methods are integrated with the unscented KF (UKF) for nonlinear model updating in this paper, they can be alternatively combined with other types of nonlinear KFs such as the extended KF (EKF) or the ensemble KF (EnKF). The performance of the proposed methods is investigated through two numerical applications and compared to that of a non-adaptive UKF and an existing dual adaptive filter. The first application considers a nonlinear steel pier where nonlinear material properties are selected as updating parameters. Significant improvements in parameter estimation results are observed when using adaptive filters compared to the non-adaptive approach. Furthermore, the covariance matrix of simulated measurement noise is estimated from the adaptive approaches with acceptable accuracy. Effects of different types of modeling errors are studied in the second numerical application of a nonlinear 3-story 3-bay steel frame structure. Similarly, more accurate and robust parameter estimations and response predictions are obtained from the adaptive approaches compared to the non-adaptive approach. The results verify the effectiveness and robustness of the proposed adaptive filters. The forgetting factor and moving window methods are shown to have a simpler tuning process compared to the dual adaptive method while providing similar performance.

Inferring kinetic parameters of oscillatory gene regulation from single cell time-series data.

Gene expression dynamics, such as stochastic oscillations and aperiodic fluctuations, have been associated with cell fate changes in multiple contexts, including development and cancer. Single cell live imaging of protein expression with endogenous reporters is widely used to observe such gene expression dynamics. However, the experimental investigation of regulatory mechanisms underlying the observed dynamics is challenging, since these mechanisms include complex interactions of multiple processes, including transcription, translation and protein degradation. Here, we present a Bayesian method to infer kinetic parameters of oscillatory gene expression regulation using an auto-negative feedback motif with delay. Specifically, we use a delay-adapted nonlinear Kalman filter within a Metropolis-adjusted Langevin algorithm to identify posterior probability distributions. Our method can be applied to time-series data on gene expression from single cells and is able to infer multiple parameters simultaneously. We apply it to published data on murine neural progenitor cells and show that it outperforms alternative methods. We further analyse how parameter uncertainty depends on the duration and time resolution of an imaging experiment, to make experimental design recommendations. This work demonstrates the utility of parameter inference on time course data from single cells and enables new studies on cell fate changes and population heterogeneity.

A Novel Robust Nonlinear Kalman Filter Based on Multivariate Laplace Distribution

In this brief, we investigate the robust state estimation of nonlinear systems with heavy-tailed noise. Based on the fact that the multivariate Laplace (ML) distribution has heavier tails than Gaussian distribution but is only determined by its mean and covariance, we utilize the ML distribution to model the heavy-tailed measurement noise. Based on the Gaussian mixture form of ML distribution, the system state and the covariance of ML distribution are inferred by variational Bayesian method, and then a closed recursive robust nonlinear Kalman filter is obtained with the nonlinear integrations calculated by numerical integration methods. Simulation results demonstrate that the proposed algorithm is insensitive to its free parameters and can obtain better estimation accuracy than related robust algorithms.

Survey Expectations and Learning

In this paper, we evaluate a model that describes real-time inflation data together with the inflation expectations measured by the Survey of Professional Forecasters (SPF). We work with a second-order autoregressive model in which the agents learn over time the intercept and persistence coefficients based on real-time data. To model the process of revisions in real time data, we allow for news and noise disturbances. In contrast to the usual time-varying parameter vector autoregression, we use non-linear Kalman filter techniques to estimate the time-varying coefficients of the underlying inflation process. We identify systematic changes in the persistence of the inflation process and in the long-run expected inflation rate that are implied by the model. The inflation forecasts implied by the model are then compared with the SPF forecasts. As we cannot reject the hypothesis that the SPF forecasts are produced based on our model, we re-estimate the model using Survey nowcasts and forecasts as additional observables. This augmented model does not change the nature and magnitude of the time variation in the coefficients of the autoregressive model, but it does help to reduce the uncertainty in the estimates. Overall, the estimated time-variation confirms our results on the perceived inflation process present in estimated DSGE models with learning (Slobodyan and Wouters, 2012a, 2012b).

Sensor fault–tolerant control of a quadrotor unmanned aerial vehicle

This article handles the issue of fault-tolerant control of a quadrotor unmanned aerial vehicle (UAV) in the existence of sensor faults. A general non-linear model of the quadrotor is presented. Several non-linear Kalman filters namely, the extended Kalman filter, the unscented Kalman filter and the cubature Kalman filter (CKF) are utilized to estimate the states of the quadrotor and to compare the estimation performances. Some flight scenarios are simulated, and the simulation results show that the CKF has the smallest estimation error as expected in theory. Control of the quadrotor heavily depends on the measured values received from sensors. Therefore, the control system requires fault-free sensors. However, small quadrotors and UAVs are mostly equipped with low-cost and low-quality sensors, and hence, they may fail to indicate correct measurement values. If the sensors are faulty, then the control system itself should be actively tolerant to sensor faults. Measurements of these kinds of sensors suffer from bias and external noise due to temperature variations, vibration and other external conditions. Since the bias is one of the very common faults in these sensors, a sensor bias is taken into consideration as a fault and occurs abruptly at a certain time and continues throughout the considered scenarios. By using the residual signals generated by the non-linear filters, sensor faults are detected and isolated. Then, two different methods are proposed for removing the effects of faults and achieving active fault–tolerant control. The effectiveness of the presented two techniques is shown in the simulations.

An Autonomous Space Navigation System Using Image Sensors

Space navigation has been a research topic for decades, with autonomy within its remit being a challenging area. Conventional space navigation approaches are based on utilizing the signal from ground facilities on the earth or using navigation sensors with sophisticated methodologies. Hence, this paper proposes a simple space autonomous navigation system that utilizes the angle measurements obtained by three image sensors. A sensitivity analysis is performed to rigorously investigate the relationship between the sensors’ performance and navigation error. A nonlinear Kalman filter is designed to improve navigation performance due to sensor errors. The performance of the proposed system is verified via nonlinear numerical simulations and Monte-Carlo simulation with sensor errors.