Nonlinear Kalman Filter Research Articles

EEG ENHANCEMENT USING EXTENDED KALMAN FILTER TO TRAIN MULTI-LAYER PERCEPTRON

In many applications of signal processing, especially in biomedicine, electroencephalogram (EEG) is the recording of electrophysiological brain activity along the scalp over a small interval of time and it is a biological non stationary signal which contains important information. Analysis of EEG signal is useful to identify physiological situations of the human as normal and epileptic subject. EEG signal becomes more complicated to be analyzed by the introduction of the noise. In this paper, a nonlinear Kalman Filter scheme where an extended Kalman filter (EKF) based Multi-layer perceptron (MLP) model is proposed to remove white and colored Gaussian noises from EEG recordings in physiological and pathological states (normal and epileptic). The MLP is one of the artificial neural network (ANN) models that has great track of impacts at solving a variety of problems. Activation function is one of the elements in MLP neural network. Selection of the activation function as sigmoid in the MLP network plays an essential role on the network performance. Thus, the MLP parameters as weights, and outputs are trained by an EKF in order to minimize the difference between the output of the neural network and the desired outputs. The results comparison studies are evaluated with root mean square difference (RMSD) and signal to noise ratio (SNR). The elapsed time is decreased using this method compared to normalised least mean square (NLMS) and Meyer wavelet methods. These parameters applied to EEG signals show the validity and effectiveness of the proposed approach.

Design of set-point weighting-based dynamic integral sliding mode control with nonlinear full-order state observers for quadcopter UAVs

This research is to develop set-point weighting-based dynamic integral sliding mode control with nonlinear full-order state observers to deal with nonlinear and underactuated coupled systems, and unforeseen circumstances of quadcopter UAVs systems. A comparative assessment through numerical simulations of sliding mode-based nonlinear observer approaches and Kalman filter is presented. These include quasi method, interval type-2 fuzzy logic system, super-twisting algorithm, higher order sliding mode observer, and extended Kalman filter. Chattering, noise rejection, estimation error and time required to track true states are evaluated to demonstrate the performance of each observer. In addition, to assess the proposed controller performance, maximum overshoot, rise time, chattering, and steady-state error are evaluated in relation to the use of each observer.

A robust damping controller for DFIG based on variable-gain sliding mode and Kalman filter disturbance observer

This paper proposes a sliding mode based inter-area oscillation damping control strategy for the Doubly Fed Induction Generator (DFIG) to enhance the stability of interconnected power systems. The synthesized damping controller is designed based on the variable-gain super-twisting algorithm to deal with the inherent nonlinear behavior in the DFIG and the power system. A Derivative-free nonlinear Kalman Filter is also developed as a disturbance estimator to accurately estimate the disturbance terms in the represented system. Such estimated results make the gains of super-twisting algorithm more efficient. The proposed controller is insensitive to the presence of unmolded system dynamics and the external disturbances acting on the system, and has a fast response speed and robustness with respect to the power system uncertainties. The stability property of the proposed damping controller is proved via the Lyapunov method, and its performance has been verified through numerical simulations considering detailed system model, external disturbances and operation point uncertainty.

Simultaneous State and Parameter Estimation using Robust Receding-horizon Nonlinear Kalman Filter

Abstract Use of measurements corrupted with gross errors for state estimation leads to biased state estimates, which, in turn, deteriorates the performances of model based process monitoring and control schemes. In this work, with the aim of minimizing the effects of gross errors on state estimates, robust versions of receding horizon nonlinear Kalman filter (RNK) are developed by integrating M-estimators with RNK. Two M-estimators, viz. Huber’s fair function and Hampel’s redescending estimator, have been considered for incorporating robustness in RNK. The proposed robust RNK formulation is further used to develop a robust simultaneous state and parameter estimation scheme. The efficacies of the proposed estimation schemes have been demonstrated by conducting simulation studies on a benchmark continuous stirred tank reactor (CSTR) system. The simulation studies reveal that, the proposed robust RNK formulations are able to generate estimation performances comparable to that of robust moving horizon estimation (MHE) scheme. Further, the proposed robust RNK based state and parameter estimator is able to estimate drifting model parameter accurately even in the presence of biased temperature measurements.

Attitude Estimation for Guided Projectiles Based on Magnetometers and Accelerometers

In order to accurately measure the attitude of guided projectile, a low cost strapdown navigation method using magnetic sensors and accelerometers is proposed. By establishing a simplified angular motion model, the state equation of nonlinear Kalman filter is obtained. According to the measurement principle and coordinate transformation of magnetic sensors and accelerometers, the measurement equation of nonlinear Kalman filter is derived. On this basis, in the simulation of the full trajectory, unscented Kalman filter (UKF) and mixed Kalman filter (MKF) are used to estimate the attitude angles. The results show that the estimation errors of the integrated system are reduced to 8.53%, 14.72% and 3.99% for angle of attack, sideslip angle and roll angle respectively, compared to the system with magnetic sensors only. Moreover, the MKF algorithm can reduce the amount of computation to 68.38% to meet the requirement of real-time performance.

State Estimation of Vehicle’s Dynamic Stability Based on the Nonlinear Kalman Filter

An accurate estimation of a vehicle’s state of motion is the basis of dynamic stability control. Two different nonlinear Kalman filters are adopted for the estimation of the vehicle’s lateral/rollover stability state. First, the overall structure of the state estimation with four inputs and four outputs is introduced. After determining tire-cornering stiffness using a recursive least-squares (RLS) method, the equations of state and of observation for the nonlinear Kalman filter are established based on a vehicle model with four degrees of freedom including planar and rollover dynamics. Then, the specific steps of real-time state estimation using the extended Kalman filter (EKF) and unscented Kalman filter (UKF) are both given. In a co-simulation, we find that the RLS algorithm estimates tire-cornering stiffness accurately and quickly, and the UKF improves the effect of state estimation compared with EKF. In addition, the UKF is verified against data from vehicle tests. The results show the proposed method is reliable and practical in estimating vehicle states.

Low-complexity online estimation for LiFePO4 battery state of charge in electric vehicles

This paper proposes a low-complexity online state of charge estimation method for LiFePO4 battery in electrical vehicles. The proposed method is able to achieve accurate state of charge with less computational efforts in comparison with the nonlinear Kalman filters, and also can provide state of health information for battery management system. According to the error analysis of equivalent circuit model with two resistance and capacitance, two proportional-integral filters are designed to compensate the errors from inaccurate state of charge and current measurements, respectively. An error dividing process is proposed to tune the contribution of each filter to the finial estimation results, which enhances the validation and accuracy of the proposed method. Recursive least squares filter can provide the state of health information and updates the parameters of battery model online to eliminate the errors caused by parameters uncertainty. The proposed method is compared with extend Kalman filter in regards to accuracy and execution time. The execution time of the proposed method is measured on Zynq board platform to validate its suitability for online implementation. In this paper, the proposed method is able to obtain less than 1% error for state of charge estimation.

Forecasting of Power Corporations’ Default Probability With Nonlinear Kalman Filtering

This paper proposes a systematic method for forecasting default probabilities for financial firms with particular interest in electric power corporations. According to the credit risk theory, a company's closeness to default is determined by the distance of its assets’ value from its debts. The assets’ value depends primarily on the company's market (option) value through a complex nonlinear relation. By forecasting with accuracy the enterprize's option value, it becomes also possible to estimate the future value of the enterprize's asset value and the associated probability of default. This paper proposes a systematic method for forecasting the probability to default for companies (option/asset value forecasting methods) using a new nonlinear Kalman filtering method under the name derivative-free nonlinear Kalman filter. The firm's option value is considered to be described by the Black–Scholes nonlinear partial differential equation (PDE). Using a differential flatness theory, the PDE is transformed into an equivalent state-space model in the so-called canonical form. Using the latter model and by redesigning the derivative-free nonlinear Kalman filter as a m -step-ahead predictor, estimates are obtained of the company's future option values. By forecasting the company's market (option) values, it becomes also possible to forecast the associated asset value and volatility, and finally, to estimate the company's future default risk.

Maximum Correntropy Unscented Kalman Filter for Ballistic Missile Navigation System based on SINS/CNS Deeply Integrated Mode

Strap-down inertial navigation system/celestial navigation system (SINS/CNS) integrated navigation is a high precision navigation technique for ballistic missiles. The traditional navigation method has a divergence in the position error. A deeply integrated mode for SINS/CNS navigation system is proposed to improve the navigation accuracy of ballistic missile. The deeply integrated navigation principle is described and the observability of the navigation system is analyzed. The nonlinearity, as well as the large outliers and the Gaussian mixture noises, often exists during the actual navigation process, leading to the divergence phenomenon of the navigation filter. The new nonlinear Kalman filter on the basis of the maximum correntropy theory and unscented transformation, named the maximum correntropy unscented Kalman filter, is deduced, and the computational complexity is analyzed. The unscented transformation is used for restricting the nonlinearity of the system equation, and the maximum correntropy theory is used to deal with the non-Gaussian noises. Finally, numerical simulation illustrates the superiority of the proposed filter compared with the traditional unscented Kalman filter. The comparison results show that the large outliers and the influence of non-Gaussian noises for SINS/CNS deeply integrated navigation is significantly reduced through the proposed filter.

EGP-CDKF for Performance Improvement of the SINS/GNSS Integrated System

Position and orientation system (POS), which typically integrates strapdown inertial navigation system (SINS) and global navigation satellite system (GNSS), serves as a key sensor in airborne remote sensing, mobile mapping, and vehicle localization. POS can provide reliable, high-frequency and high-precision motion parameters using nonlinear Kalman Filter models based on fusion methods, such as extended Kalman filter, unscented Kalman filter, central difference Kalman filter (CDKF), and square root CDKF (SR-CDKF). Although the nonlinear parametric models are of high efficiency, there are also limitation on their capabilities of prediction and estimation, as it is often impossible to model all aspects of POS. In this paper, Gaussian processes (GP)-based is presented to enhance the capabilities of prediction and estimation for parametric CDKF. On one hand, it can estimate the state vector of POS with the nonlinear parametric CDKF on condition that trained data is limited; on the other hand, GP can take both the noise and the uncertainty in the nonlinear parametric CDKF into consideration. Consequently, the incorporation of GP into CDKF can result in further performance improvement. The proposed approach is verified in the real experiment, and shows that large performance benefits are achieved through applying the enhanced GP-CDKF(EGP-CDKF) into the SINS/GNSS integrated system.

Estimating traffic conditions from smart work zone systems

ABSTRACTThis article evaluates the effectiveness of sensor network systems for work zone traffic estimation. The comparative analysis is performed on a work zone modeled in microsimulation and calibrated with field data from an Illinois work zone. Realistic error models are used to generate noisy measurements corresponding to Doppler radar sensors, remote traffic microwave sensors (RTMSs), and low energy radars. The velocity, queue length, and travel time are estimated with three algorithms based on (i) interpolation, (ii) spatio-temporal smoothing, and (iii) a flow model–based Kalman filter. A total of 396 sensor and algorithm configurations are evaluated and the accuracy of the resulting traffic estimates is compared to the true traffic state from the microsimulation. The nonlinear Kalman filter provides up to 30% error reduction over other velocity estimators when the RTMS sensor spacing exceeds two miles, and generally offers the best performance for queue and travel time estimation.

Broadband Load Torque Estimation in Mechatronic Powertrains Using Nonlinear Kalman Filtering

An important bottleneck in the design, operation, and exploitation of mechatronic powertrains is the lack of accurate knowledge of broadband external loading. This is caused by the intrusive nature of regular torque measurements. This paper proposes a novel nonintrusive approach to obtain torsional load information on mechatronic powertrains. Online coupled state/input estimation is performed through an augmented nonlinear Kalman filter. This estimation approach exploits general lumped-parameter physics-based models in order to create a widely applicable framework. This paper considers both extended (EKF) and unscented Kalman filtering approaches. Contrary to previous works, no considerable difference in accuracy is obtained from experiments, with a considerably lower computational load for the EKF. This paper reveals the benefits of including rotational acceleration measurements from a theoretical perspective, which is demonstrated through the experimental validation. This drastically increases the broadband accuracy. The result of this paper is an accurate and noninvasive virtual torque sensor with a sufficiently broad bandwidth for use in condition monitoring, control, and future design optimization.

Convergence analysis of nonlinear Kalman filters with novel innovation-based method

Abstract The convergence of nonlinear Kalman filters has conventionally been analyzed in terms of the estimation error. In this paper, we present a new method for investigating the convergence performance of a class of nonlinear Kalman filters based on deterministic sampling. The systems considered here are described by nonlinear state equations with linear measurements. For this type of systems, our proposed convergence analysis is performed using “innovation”, which is defined as the error between the measurement and its prediction. Specifically, we obtain a linear relationship between the innovation and the estimation error, and derive a set of sufficient conditions that ensures the convergence of nonlinear Kalman filters. Compared with the conventional convergence analysis method based on the estimation error, the proposed innovation-based method can obtain sufficient conditions for convergence more directly and readily. Simulation results show that the convergence of innovation generates the convergence of nonlinear Kalman filters.

Nonlinear Kalman filters for aircraft engine gas path health estimation with measurement uncertainty

This paper is concerned with nonlinear Kalman filtering approach to aircraft engine gas path analysis with measurement uncertainty. The uncertain measurements are characterized by time delay and packet dropout. The delay step of physical parameters occurs randomly, and its probability is regulated by a set of uncorrelated variables following Poisson distribution and uniform distribution. Packet dropout is caused as the data are not collected in time or data buffer overflows. The novel nonlinear Kalman filters (KFs) are developed using a multistep recursive estimation strategy with self-tuning buffer in the presence of gas path measurement uncertainty. The data buffers are introduced in the state estimator, the length of which is adaptive to the information loss level. The algorithms run recursively using the new arrival data and buffer position information. With a more effective arrangement of the collected measurements in real time, the better estimation accuracy of gas path health status is expected. Simulations involving abrupt fault and degradation datasets of aircraft engines were carried out to numerically evaluate and compare the performance of the improved nonlinear KFs with their existing KFs in the context of health estimation with time delay and packet dropout. The test results demonstrate that the proposed methodology not only reduces the computational time but also obtains a satisfactory accuracy for state estimation in the cases of engine gas path measurement uncertainty.

Phase unwrapping algorithm using polynomial phase approximation and linear Kalman filter.

A noise-robust phase unwrapping algorithm is proposed based on state space analysis and polynomial phase approximation using wrapped phase measurement. The true phase is approximated as a two-dimensional first order polynomial function within a small sized window around each pixel. The estimates of polynomial coefficients provide the measurement of phase and local fringe frequencies. A state space representation of spatial phase evolution and the wrapped phase measurement is considered with the state vector consisting of polynomial coefficients as its elements. Instead of using the traditional nonlinear Kalman filter for the purpose of state estimation, we propose to use the linear Kalman filter operating directly with the wrapped phase measurement. The adaptive window width is selected at each pixel based on the local fringe density to strike a balance between the computation time and the noise robustness. In order to retrieve the unwrapped phase, either a line-scanning approach or a quality guided strategy of pixel selection is used depending on the underlying continuous or discontinuous phase distribution, respectively. Simulation and experimental results are provided to demonstrate the applicability of the proposed method.

Simultaneous State and Parameter Estimation using Receding-horizon Nonlinear Kalman Filter

Online estimation of internal states and parameters is often required for process monitoring, control and fault diagnosis. The conventional approach to estimate drifting parameters is to artificially model them 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, Valluru et al. (2017) have developed a moving window based state and parameter estimator which assumes that the parameters change slowly and remain constant within the window. Also, in another development, a moving window based recursive filter, receding horizon nonlinear Kalman (RNK) filter has been proposed by Rengaswamy et al. (2013). In this work, a novel simultaneous state and parameter estimator is proposed by combining the window based parameter variation model with RNK filter formulation. The performance of the RNK based estimator is demonstrated by conducting simulation studies on the benchmark quadruple tank system and a CSTR system. The efficacy of RNK based estimator is compared with that of the conventional simultaneous EKF approach and Moving Horizon Estimator (MHE) based state and parameter approach. Analysis of the simulation results reveals that the proposed state and parameter estimation scheme is able to generate better estimation performance than that of the simultaneous EKF and closer to that of the MHE based parameter estimator with less computational efforts.

Online Parameter Identification and State of Charge Estimation of Lithium-Ion Batteries Based on Forgetting Factor Recursive Least Squares and Nonlinear Kalman Filter

State of charge (SOC) estimation is the core of any battery management system. Most closed-loop SOC estimation algorithms are based on the equivalent circuit model with fixed parameters. However, the parameters of the equivalent circuit model will change as temperature or SOC changes, resulting in reduced SOC estimation accuracy. In this paper, two SOC estimation algorithms with online parameter identification are proposed to solve this problem based on forgetting factor recursive least squares (FFRLS) and nonlinear Kalman filter. The parameters of a Thevenin model are constantly updated by FFRLS. The nonlinear Kalman filter is used to perform the recursive operation to estimate SOC. Experiments in variable temperature environments verify the effectiveness of the proposed algorithms. A combination of four driving cycles is loaded on lithium-ion batteries to test the adaptability of the approaches to different working conditions. Under certain conditions, the average error of the SOC estimation dropped from 5.6% to 1.1% after adding the online parameters identification, showing that the estimation accuracy of proposed algorithms is greatly improved. Besides, simulated measurement noise is added to the test data to prove the robustness of the algorithms.

ADAPTIVE UNSCENTED KALMAN FILTER FOR ONLINE SOFT TISSUES CHARACTERIZATION

Online soft tissue characterization is important for robotic-assisted minimally invasive surgery to achieve precise and stable robotic control with haptic feedback. This paper presents a new adaptive unscented Kalman filter based on the nonlinear Hunt–Crossley model for online soft tissue characterization without requiring the characteristics of system noise. This filter incorporates the concept of Sage windowing in the traditional unscented Kalman filter to adaptively estimate system noise covariance using predicted residuals within a time window. In order to account for the inherent relationship between the current and previous states of soft tissue deformation involved in robotic-assisted surgery and improve the estimation performance, a recursive estimation of system noise covariance is further constructed by introducing a fading scaling factor to control the contributions between noise covariance estimations at current and previous time points. The proposed adaptive unscented Kalman filter overcomes the limitation of the traditional unscented Kalman filter in requiring the characteristics of system noise. Simulations and comparisons show the efficacy of the suggested nonlinear adaptive unscented Kalman filter for online soft tissue characterization.

A Novel Methodology for Estimating State-Of-Charge of Li-Ion Batteries Using Advanced Parameters Estimation

State-of-charge (SOC) estimations of Li-ion batteries have been the focus of many research studies in previous years. Many articles discussed the dynamic model’s parameters estimation of the Li-ion battery, where the fixed forgetting factor recursive least square estimation methodology is employed. However, the change rate of each parameter to reach the true value is not taken into consideration, which may tend to poor estimation. This article discusses this issue, and proposes two solutions to solve it. The first solution is the usage of a variable forgetting factor instead of a fixed one, while the second solution is defining a vector of forgetting factors, which means one factor for each parameter. After parameters estimation, a new idea is proposed to estimate state-of-charge (SOC) of the Li-ion battery based on Newton’s method. Also, the error percentage and computational cost are discussed and compared with that of nonlinear Kalman filters. This methodology is applied on a 36 V 30 A Li-ion pack to validate this idea.

A Study of Unscented Kalman Filter Performance in the Ultralight Coupled Integration System

This paper aims to the enhancement of the performance in the navigation filter of the GNSS/INS ultralight coupled (UTC) integration system. The basic structure of the UTC and the state-space equations are provided in this paper. Owing to its strong nonlinearity, a kind of nonlinear Kalman filter based on the unscented transformation is introduced to the system, which can lead to higher accuracy and stronger robustness than traditional EKF. Furthermore, when applying into a realistic nonlinear system without detailed information about the model and noise, an advanced UKF is proposed in this paper to deal with the potential increasing state-error and divergence of the system. Finally, Simulations are made to prove the availability of these theories above.