Nonlinear State Estimation Methods Research Articles

Detecting Weak Physical Signal from Noise: A Machine-Learning Approach with Applications to Magnetic-Anomaly-Guided Navigation

Detecting a weak physical signal immersed in overwhelming noises entails separating the two, a task for which machine learning is naturally suited. In principle, such a signal is generated by a nonlinear dynamical system of intrinsically high dimension for which a mathematical model is not available, rendering unsuitable traditional linear or nonlinear state-estimation methods that require an accurate system model (e.g., extended Kalman filters). We exploit the architectures of reservoir computing and feed-forward neural networks (FNNs) with time-delayed inputs to solve the weak-signal-detection problem. As a prototypical example, we apply the machine-learning schemes to Earth's magnetic anomaly field-based navigation. In particular, the time series are collected from the interior of the cockpit of a flying aircraft during different maneuvering phases, where the overwhelmingly strong noise background is the result of other components of Earth's magnetic field and the fields generated by the electronic devices in the cockpit. We demonstrate that, when combined with the traditional Tolles-Lawson model for Earth's magnetic field, the articulated machine-learning schemes are effective for accurately detecting the weak anomaly field from the noisy time series. The schemes can be applied to detecting weak signals in other domains of science and engineering.

The Simple Solution for Nonlinear State Estimation of Ill-Conditioned Systems: The Normalized Unscented Kalman Filter

An easy-to-implement method for nonlinear state estimation for ill-conditioned systems is proposed. By propagating standard deviations and correlations instead of the covariance in the unscented Kalman filter (UKF), the condition numbers of relevant matrices are reduced. The reduction in the condition number is related to the scaling of the problem. Hence, what we propose is a normalization method that acts as an “auto-scaler”. Compared to other methods in state estimation for ill-conditioned systems, our proposed method factors the covariance matrix into physically meaningful statistics which can be used to check for filter divergence online. The method is compared to a standard UKF in a case study and shows a significant reduction in the condition number.

State Estimation of Gas-Lifted Oil Well Using Nonlinear Filters.

The focus of this work is the extension of nonlinear state estimation methods to gas-lifted systems. The extended Kalman filter (EKF), unscented Kalman filter (UKF) and particle filter (PF) were used to estimate the nonlinear states. Brief descriptions of the filters were first presented starting from the linear Kalman filter. Hypothesis tests on the expectation of the residuals were performed to show how close to optimal the estimation methods are and it showed the UKF estimates to be slightly better than EKF while PF performs the worst. The PF has poor accuracy using residual visualisation, hypothesis test and the root mean squared error (RMSE) values of the residuals. The gas-lifted system exhibits casing heading instability where the states show oscillatory behaviour depending on the value of the input but the results here do not change in a known way for each filter as the input is changed from the non-oscillatory region to the oscillatory region. Therefore, for this noise distribution and model assumption, either the EKF or UKF can be used for nonlinear state estimation with UKF better preferred if computational cost is not considered when control solutions are used in gas-lifted system.

Interval Extended Kalman Filter—Application to Underwater Localization and Control

The extended Kalman filter has been shown to be a precise method for nonlinear state estimation and is the facto standard in navigation systems. However, if the initial estimated state is far from the true one, the filter may diverge, mainly due to an inconsistent linearization. Moreover, interval filters guarantee a robust and reliable, yet unprecise and discontinuous localization. This paper proposes to choose a point estimated by an interval method, as a linearization point of the extended Kalman filter. We will show that this combination allows us to get a higher level of integrity of the extended Kalman filter.

Research on Fault Detection Method of Wind Turbine Generator Based on SCADA Data

In this paper, a data driven fault detection model of the generator based on improved nonlinear state estimation (INSET) method was established and the residuals of all the relevant parameters were predicted synchronously. According to the residual distribution characteristics the alarm rules were designed. In case studies, the fault was detected timely and exactly for avoiding serious accident. In addition, compared with other conventional algorithms, the results showed higher prediction accuracy and sensitivity and indicated the feasibility of the method.

Random Weighting-Based Nonlinear Gaussian Filtering

The Gaussian filtering is a commonly used method for nonlinear system state estimation. However, this method requires both system process noise and measurement noise to be white noise sequences with known statistical characteristics. However, it is difficult to satisfy this condition in engineering practice, making the Gaussian filtering solution deviated or diverged. This paper adopts the random weighting concept to address the limitation of the nonlinear Gaussian filtering. It establishes the random weighting estimations of system noise characteristics on the basis of the maximum a-posterior theory, and further develops a new Gaussian filtering method based on the random weighting estimations to restrain system noise influences on system state estimation by adaptively adjusting the random weights of system noise characteristics. Simulation, experimental and comparison analyses prove that the proposed method overcomes the limitation of the traditional Gaussian filtering in requirement of system noise characteristics, leading to improved estimation accuracy.

Optimal Reduction of Dirac Mixture Densities on the 2-Sphere

This paper is concerned with optimal approximation of a given Dirac mixture density on the S2 manifold, i.e., a set of weighted samples located on the unit sphere, by an equally weighted Dirac mixture with a reduced number of components. The sample locations of the approximating density are calculated by minimizing a smooth global distance measure, a generalization of the well-known Cramér-von Mises Distance. First, the Localized Cumulative Distribution (LCD) together with the von Mises–Fisher kernel provides a continuous characterization of Dirac mixtures on the S2 manifold. Second, the L2 norm of the difference of two LCDs is a unique and symmetric distance between the corresponding Dirac mixtures. Thereby we integrate over all possible kernel sizes instead of choosing one specific kernel size. The resulting approximation method facilitates various efficient nonlinear sample-based state estimation methods.

Approximating Sample State Vectors Using the ESPT for Computationally Efficient Particle Filtering

The extrapolated single propagation technique (ESPT) based particle filter is proposed. Instead of propagating all the sample state vectors using multiple numerical integrations, only one sample state vector is propagated in the ESPT. Other sample states are computed at the next epoch with an accuracy of the second-order Taylor series expansion terms. It is theoretically shown that the particle filter algorithm, which uses the ESPT for state propagation, requires up to $\text{90}\%$ reduced processing time than the conventional particle filter. Using a benchmark non-linear estimation problem the performance of the modified algorithm is compared with the particle filter. The experimental results confirm the theoretical analysis and demonstrate the effectiveness of the proposed method for non-linear state estimation. The proposed algorithm significantly reduces the processing time of the particle filter with a minor degradation in the estimation accuracy.

Adaptively Random Weighted Cubature Kalman Filter for Nonlinear Systems

This paper presents a new adaptive random weighting cubature Kalman filtering method for nonlinear state estimation. This method adopts the concept of random weighting to address the problem that the cubature Kalman filter (CKF) performance is sensitive to system noise. It establishes random weighting theories to estimate system noise statistics and predicted state and measurement together with their associated covariances. Subsequently, it adaptively adjusts the weights of cubature points based on the random weighting estimations to improve the prediction accuracy, thus restraining the disturbances of system noises on state estimation. Simulations and comparison analysis demonstrate the improved performance of the proposed method for nonlinear state estimation.

Using smart meters for checking the topology and power flow calculation of a secondary distribution network

The paper analyzes options of using smart meters for power flow calculation and for assessing the state of a real three-phase four-wire secondary distribution network based on measurements of average values of active and reactive power and of voltages. The work is based on the authors’ research on allocation of measurements to ensure secondary distribution network observability and on selection of the most efficient method for linear and non-linear state estimation. The paper illustrates solution of a problem on identification composition of load nodes in the phases and reveals challenges related to voltage account in the neutral wire and in its grounding.

Nonlinear state estimation with delayed measurements using data fusion technique and cubature Kalman filter for chemical processes

Nonlinear state estimation with delayed measurements has been considered in many industrial applications. However, classical methods cannot use these slow rates, irregular, delayed measurements, even though the delayed measurements are usually more accurate. Therefore, finding a method to utilize these delayed measurements can improve the accuracy and robustness of nonlinear state estimation. As this aim, one nonlinear state estimation method with delayed measurements using data fusion technique and cubature Kalman filter is proposed. The framework of processing delayed measurements was elaborated by applying the data fusion technique of covariance matrix. Then, two kinds of data fusion methods, with corresponding merits and faults in speed and accuracy, were described. Finally, the efficacy of the proposed methods is demonstrated by a chemical application of the nonlinear polymerization process.

Incorporating delayed measurements in an improved high-degree cubature Kalman filter for the nonlinear state estimation of chemical processes

The on-line estimation of process quality variables has a large impact on the advanced monitoring and control techniques of chemical processes. The present study offers an improved high-degree cubature Kalman filter (HCKF) to solve the nonlinear state estimation problem of high-dimensional chemical processes. We substituted the Cholesky decomposition in the HCKF filter with a diagonalization transformation of the matrix. In addition, we enhanced numerical stability and estimation accuracy. On this basis, we present one nonlinear state estimation method based on the sample-state augmentation and improved HCKF to handle issues with delayed measurements. Finally, we used the nonlinear state estimation experiments for the polymerization process to validate the proposed method. The numerical results indicated the achievement of state estimation with higher accuracy and better stability following the effective utilization of the delayed measurements for nonlinear chemical processes.

Nonlinear state estimation under bounded noises

Most of the existing nonlinear state estimation methods require to know the statistical information of noises. However, the statistical information may not be accurately obtained or satisfied in practical applications. Actually, the noises are always bounded in a practical system. In this paper, we study the nonlinear state estimation problem under bounded noises, where the addressed noises do not provide any statistical information, and the bounds of noises are also unknown. By using matrix analysis and second-order Taylor series expansion, a novel constructive method is proposed to find an upper bound of the square error of the nonlinear estimator. Then, a convex optimization problem on the design of an optimal estimator gain is established in terms of linear matrix inequalities, which can be solved by standard software packages. Moreover, stability conditions are derived such that the square error of the designed nonlinear estimator is asymptotically bounded. Finally, two illustrative examples are employed to show the advantages and effectiveness of the proposed methods.

Enhancement of chaotic hydrological time series prediction with real-time noise reduction using Extended Kalman Filter

Short-term prediction of hydrological time series using chaotic dynamical systems approach has gained popularity. However, noise can severely affect the prediction accuracy of any data driven modelling approach. Noise reduction processes attempted in chaotic hydrological time series analysis were later questioned by Elshorbagy et al. (2002) for their appropriateness for noise reduction processes. Only a few studies on noise reduction have appeared since then, and these too were limited to introducing new noise reduction methods.Most of the past studies on chaotic time series noise reduction shared a general perception that the noise reduced data trained models can give better predictions; they used off line noise reduction methods to obtain a ‘noise reduced data set’ which was then used to make prediction models. However, the results of this study showed that this was not necessarily true and often flawed, when the input data was noisier. Instead, noise-reduced data inputs, were shown to yield higher prediction accuracy even with noisy data trained models. Thus, the study showed a real need to use real-time noise reduction methods instead of off-line noise reduction methods that had previously been used.The Extended Kalman Filter (EKF), a popular nonlinear state estimation method from controls literature, was introduced as a real-time noise reduction method and its effectiveness was demonstrated on both synthetic chaotic time series and real river flow time series. EKF produced prediction improvement as high as 15%–40% on the benchmark time series with noise levels varying from 1% to 30%. Two river flow series, with low average flows, showed prediction improvement whereas three other flow series, with relatively large average flows, did not. Artificial Neural Network (ANN) models were used as the state-space models in EKF, and adopting them to time delays different from 1 unit was also demonstrated. The study demonstrated an ‘indirect’ validation method to verify the effectiveness of noise reduction when several interrelated time series were available; this was supported in observed discharge time series of the Ciliwung River in Jakarta, Indonesia.

Improved model based fault detection technique and application to humanoid robots

One of the issue of significant interest for robotics is the fault detection, specifically when we have application in risky circumstances. Robotic systems required a capacity to efficiently identify and endure some defects so that they can keep achieving the required tasks while avoiding instantaneous repairing process. Consequently, we aim in this work to propose a systematic approach for state estimation and fault detection technique to enhance the operation of humanoid robots (HR) systems using an extended Kalman filter (EKF)-based multiscale optimized exponentially weighted moving average chart (MS-OEWMA). The objectives of this work are sixfold: (1) apply EKF technique to estimate the state variables in HR systems. The EKF is among the most popular nonlinear state estimation methods; (2) use dynamical multiscale representation for obtaining accurate settled characteristics; (3) propose a new optimized EWMA (OEWMA) based on the best selection of both smoothing parameter (λ) and control width L; (4) combine the advantages of state estimation technique with MS-OEWMA chart to improve the monitoring of HR systems; (5) investigate the effect of fault types (change in variance and mean in shift) and fault sizes on the monitoring performances; (6) validate the developed technique using two robot models: inverted pendulum and five-bar linkage. The detection results are evaluated using three fault detection metrics: missed detection rate (MDR), false alarm rate (FAR) and out-of-control average run length (ARL1).

Visual Odometry and Place Recognition Fusion for Vehicle Position Tracking in Urban Environments

In this paper, we address the problem of vehicle localization in urban environments. We rely on visual odometry, calculating the incremental motion, to track the position of the vehicle and on place recognition to correct the accumulated drift of visual odometry, whenever a location is recognized. The algorithm used as a place recognition module is SeqSLAM, addressing challenging environments and achieving quite remarkable results. Specifically, we perform the long-term navigation of a vehicle based on the fusion of visual odometry and SeqSLAM. The template library for this latter is created online using navigation information from the visual odometry module. That is, when a location is recognized, the corresponding information is used as an observation of the filter. The fusion is done using the EKF and the UKF, the well-known nonlinear state estimation methods, to assess the superior alternative. The algorithm is evaluated using the KITTI dataset and the results show the reduction of the navigation errors by loop-closure detection. The overall position error of visual odometery with SeqSLAM is 0.22% of the trajectory, which is much smaller than the navigation errors of visual odometery alone 0.45%. In addition, despite the superiority of the UKF in a variety of estimation problems, our results indicate that the UKF performs as efficiently as the EKF at the expense of an additional computational overhead. This leads to the conclusion that the EKF is a better choice for fusing visual odometry and SeqSlam in a long-term navigation context.

Constrained State Estimation via Projection based Optimized Parameters UKF

The unscented Kalman filter (UKF) has become a popular method for nonlinear state estimation during the last decade. However, the conventional UKF may not be suitable for real-world applications with state constrains that stem from physical definitions, physical laws or model restrictions. A UKF based method with optimized parameters was proposed in this paper to handle state constraints via the projection of sigma points. In the proposed method, the generated sigma points that violate the state constraints were projected onto the constraint boundary first. The three free parameters of the UKF, i.e., α ,β ,κ , were then optimized using a Gaussian process optimization (GPO) method. Simulations indicate that the proposed optimized UKF algorithm with the projection of sigma points can handle constrained state estimation problem effectively and efficiently.

Comparison of Unscented and Extended Kalman Filters with Application in Vehicle Navigation

The Unscented Kalman Filter (UKF) is a well-known nonlinear state estimation method. It shows superior performance at nonlinear estimation compared to the Extended Kalman Filter (EKF). This paper is devoted to an investigation between UKF and EKF with different feedback control modes in vehicle navigation. Theoretical formulation, simulation and field tests have been carried out to compare the performance of UKF and EKF. The simulation and test results demonstrate that the estimated state of a UKF relies on the measurements and is less sensitive to historical model information. The results also indicate that UKF has benefits for prototype model design due to avoidance of calculation of a Jacobian matrix. EKF, however, is more computationally efficient and more stable.

An improved extended Kalman filter with inequality constraints for gas turbine engine health monitoring

Various Kalman filter approaches have been proposed for the state estimation of gas turbine engines, among which Linear Kalman Filter (LKF) is the most common one. Kalman filters achieve state estimation provided that there are more available measurement sensors than state parameters to be estimated. However, it is hard to hold this assumption in gas turbine engine health monitoring applications, and an underdetermined estimation problem rises up.The aim of this contribution is to present a nonlinear underdetermined state estimation method on the basis of Extended Kalman Filter (EKF); and to evaluate the performance of this methodology, the comparisons of three nonlinear estimators, i.e. basic EKF, underdetermined EKF and resultant EKF are conducted to gas turbine engine health state estimation. The underdetermined EKF is developed from the previous linear achievements using the transformation matrix, and it produces the least estimation errors in the nonlinear framework. Moreover, the prior state information represented by inequality constraints is introduced to create the resultant EKF, and the estimates of state variables are tuned to truncated Probability Density Function (PDF). Results from the application to a turbojet engine health monitoring in the flight envelope show that the proposed methodology yields a significant improvement in terms of underdetermined estimation accuracy and robustness.

Nonlinear state estimation for fermentation process using cubature Kalman filter to incorporate delayed measurements

State estimation of biological process variables directly influences the performance of on-line monitoring and optimal control for fermentation process. A novel nonlinear state estimation method for fermentation process is proposed using cubature Kalman filter (CKF) to incorporate delayed measurements. The square-root version of CKF (SCKF) algorithm is given and the system with delayed measurements is described. On this basis, the sample-state augmentation method for the SCKF algorithm is provided and the implementation of the proposed algorithm is constructed. Then a nonlinear state space model for fermentation process is established and the SCKF algorithm incorporating delayed measurements based on fermentation process model is presented to implement the nonlinear state estimation. Finally, the proposed nonlinear state estimation methodology is applied to the state estimation for penicillin and industrial yeast fermentation processes. The simulation results show that the on-line state estimation for fermentation process can be achieved by the proposed method with higher estimation accuracy and better stability.