Divided Difference Filter Research Articles

Огляд методів параметричного діагностування агрегатів гідравлічних і паливних систем літальних апаратів

The subject of study in this article is the methods of parametric diagnostics of the technical condition of the units of an air vehicle's hydraulic and fuel systems, which make it possible to ensure the established reliability indicators of these systems throughout their entire life cycle. This work aims to analyze the existing literature on the methods of parametric diagnostics of hydraulic units, determine the basic requirements for the development of diagnostic methods for units of hydraulic, and fuel systems. The objective of the study is to classify the existing methods of parametric diagnostics, applied primarily to hydraulic systems, to analyze the advantages and disadvantages of the algorithms under consideration based on the requirements for diagnostic algorithms established by the authors, and to assess the possibility of effectively using these methodologies to diagnose the technical condition of a complex hydraulic distribution unit. The authors of this work formed the main requirements for the methods of diagnosing a hydraulic and fuel system's units based on the nature of the detected defects, the experience of using diagnostic algorithms, as well as the requirements of existing standards and scientific publications analysis. The authors developed a classification of parametric diagnostic algorithms to systematize existing works. This paper describes the main features and differences between diagnostic algorithms based on methods for identifying a mathematical model of an object and diagnostic algorithms in the space of measured parameters. Methods for forming diagnostic models of hydraulic units have been analyzed, such as the state-space model, the Hammerstein model, the Volterra model, the ARX model, and the matrix of influence coefficients. Established analysis of the application of defect identification algorithms such as a divided difference filter (DDF), a radial basis functions neural network RBF, and a cosine distance method. As a result, the advantages and disadvantages of the monitored diagnostic algorithms were identified and the main tasks for developing an algorithm for parametric diagnostics of the technical condition of a complex hydraulic distribution unit were formulated.

Mars entry navigation under biases based on adaptive Huber divided difference filter

SummaryTo address the problems of unknown disturbances and measurement outliers at Mars entry phase, three integrated navigation schemes are proposed in this study by complying with divided difference filters, that is, integrated navigation schemes based on the Huber first‐order divided difference filter (HDDF1), adaptive Huber first‐order divided difference filter (AHDDF1), and adaptive Huber second‐order divided difference filter (AHDDF2). The novel versions of predictive state estimation error covariance (PSEEC) and measurement noise covariance (MNC) are deduced by adopting the function of the Huber case to modify the cost function of the standard Kalman filter. To be specific, the HDDF1 is derived by embedding the novel PSEEC and MNC in the general first‐order divided difference filter. To more specifically enhance the performance of the filter under significant biases, an adaptive forgetting factor is introduced in the HDDF1. On that basis, the AHDDF1 is derived. Likewise, the AHDDF2 is yielded. The numerical simulations are presented to verify the effectiveness of the proposed navigation schemes.

Robust Divided Differences Gaussian Mixture Non Linear Filtering Applied to INS/GNSS Integrated Navigation System with Impulsive Measurement Noise

In this paper, robust INS/GNSS is designed to solve specific problem of non linear time variant state space estimation with non Gaussian measurement noise based on parallel divided difference filtering. Non linear approximation techniques such as Extended Kalman filter EKF, Interpolation Non Linear Filters called DD1 and DD2 are computed. Great comparisons are conduced and criticized in order to confirm and compare some different robust techniques explored in literature. The most tested noise in robust control and estimation theory has been simulated and disturbed GPS+GLONASS signals. Three parallel modified algorithms are simulated and compared with robust and modified forms of the same algorithms based on Root Mean Square Error RMSE Criteria.

A multi‐tone central divided difference frequency tracker with adaptive process noise covariance tuning

SummaryThe problem of real‐time frequency estimation of nonstationary multi‐harmonic signals is important in many applications. In this paper, we propose a novel multi‐frequency tracker based on a state‐space representation of the signal with Cartesian filters and the second‐order central divided difference filter (CDDF), which improves the performance of the extended Kalman filter (EKF) by using Stirling's interpolation method to approximate the mean and covariance of the state vector. A crucial element of the method is the adaptive scaling of the process noise covariance matrix appearing in the filter equations, as a function of the innovation sequence, which tunes the accuracy‐reactivity trade‐off of the filter. The proposed solution is evaluated against two approaches from the literature, namely the factorized adaptive notch filter (FANF) and the extended Kalman filter frequency tracker (EKFFT). Several experiments emphasize the estimation accuracy of the proposed method as well as the improved robustness with respect to initial errors and input signal complexity. The presented method appears to be particularly efficient with rapidly varying frequencies, thanks to the update mechanism that adjusts the filter parameters based on the amplitude of the estimation error.

Differential Evolution Optimized a Second-Order Divided Difference Particle Filter

In order to improve the estimation accuracy of particle filter algorithm in a nonlinear system state estimation problem, a new algorithm based on the second-order divided difference filter to generate the proposed distribution and the differential evolution algorithm for resampling is proposed. The second-order divided difference based on Strling’s interpolation formula is used to generate approximations to nonlinear dynamics, which avoids the evaluation of the Jacobian derivative matrix and is easy to implement. Cholesky factorization is used to ensure the positive definiteness of the covariance matrix. The truncated errors of the local linearization are reduced to a certain extent, and the approximation degree of the proposed distribution to the posterior probability of the system state is improved. The differential evolution algorithm is used to replace the traditional resampling algorithm, which effectively mitigates the problem of particle degradation. Monte Carlo simulation experiments show the effectiveness of the new algorithm.

Robust adaptive divided difference filter based on forgetting factors and its applications

SummaryThis paper proposes robust and adaptive divided difference filters (RADDFs) based on forgetting factors which are robust to dynamic systems with biases or uncertainties. The RADDFs are founded on the principle of covariance matching. The robustness of RADDFs is reflected in that it amplifies the innovation covariance to compensate the effect of dynamic biases or uncertainties. The forgetting factor is adjusted adaptively. Then, the scalar forgetting factor is further extended to multiple forgetting factors. The proposed RADDFs are illustrated by Mars entry navigation system with atmospheric density uncertainty, lift over drag ratio uncertainty, and ballistic coefficient uncertainty. To validate the filter performance by multiple forgetting factors, a typical tight coupling nonlinear system with abrupt biases is used.

Adaptive two-filter smoothing based on second-order divided difference filter for distributed position and orientation system

The distributed position and orientation system (POS) uses transfer alignment to accurately measure multi-node time-spatial reference information, which is necessary for compensating the motion error of imaging sensors used in aerial surveys. Lever-arm deformation in transfer alignment will result in time-varying covariance of measurement noise, which decreases the estimation accuracy. To address this problem, a second-order adaptive divided difference two-filter smoother (ADDTFS) is proposed for transfer alignment. The second-order adaptive divided difference filter (ADDF2) severs as the forward filter to address nonlinearity in the system. The ADDF2 can estimate the measurement noise covariance in real time based on adaptive estimation using innovation information. The weighted statistical linear regression formulation is used in the backward filter to independently estimate the states and error covariance matrices. Then the forward and backward results are combined, to obtain high-precision estimation results in post-processing. Finally, the vehicle experiment results show that the proposed method can improve the attitude precision of transfer alignment in a distributed POS by 27.84%, which has been successfully applied to motion compensation of airborne interferometric synthetic aperture radar (InSAR).

Formation of a Satellite Navigation System Using X-Ray Pulsars

The ephemeris of a pulsar is stable for a long time, which allows navigation based on pulsar orientation to be vector feasible. The formation of a satellite navigation model using the orientation vector of an X-ray pulsar signal is presented in this paper. To obtain the time difference of arrival (TDOA), a new estimation method is constructed, which can measure the photon sequence of an X-ray pulsar signal and is based on the fast Fourier transform (FFT). Next, three new observation variables are constructed. The variables are satellite phase incrementation; the angle between the satellite baseline and the pulsar direction vector; and the angle between the plane spanned by three satellite baselines and the pulsar direction vector. All three variables, along with the TDOA of the X-ray pulsar signal, are utilized to determine the orbit. The position and velocity of the satellite formations are estimated by using the adaptive divided difference filter (ADDF) to eliminate nonlinearity. Several simulation cases are designed to verify the proposed method.

A nonlinear two-filter smoothing estimation method based on DD2 filter for land vehicle POS

Land vehicle position and orientation systems (POSs) are playing an important role in ground surveying. In order to solve the problem of nonlinear error and obtain high-precision motion parameters in post-processing, a nonlinear two-filter smoothing estimation method based on second-order divided difference filter (DD2) for land vehicle POS is proposed. The DD2, based on Stirling interpolation, is applied to solve the nonlinear problem in the forward filter. The information filter is adopted to independently estimate the states and error covariance matrices in the backward filter. Then the forward and backward results are combined, which makes full use of all measurement information and also uses more model information. Therefore, more accurate estimation results for the POS can be obtained. Finally, an experiment is carried out, and the results show that the attitude precision of a land vehicle POS can be effectively improved through the proposed method.

A Nonlinear Transfer Alignment of Distributed POS Based on Adaptive Second-Order Divided Difference Filter

Distributed position and orientation system (POS) uses transfer alignment to accurately measure multi-node time-spatial reference information, which is urgently demanded to compensate motion error of imaging sensors for aerial survey. The lever-arm deformation in transfer alignment will result in nonlinear error and time-varying measurement noise covariance, which decreases the accuracy of real-time estimation. In order to solve this problem, a nonlinear transfer alignment method of distributed POS based on an adaptive second-order divided difference filter (ADDF2) is proposed. First, a transfer alignment model for distributed POS is established. Furthermore, an ADDF2 based on adaptive innovation estimation is proposed to solve the nonlinear problem and adaptively adjust the measurement noise covariance matrix of transfer alignment in real time. The flight experiment results show that the proposed method is effective to improve the precision of distributed POS.

Correntropy Based Divided Difference Filtering for the Positioning of Ships.

In this paper, robust first and second-order divided difference filtering algorithms based on correntropy are proposed, which not only retain the advantages of divided difference filters, but also exhibit robustness in the presence of non-Gaussian noises, especially when the measurements are contaminated by heavy-tailed noises. The proposed filters are then applied to the problem of ship positioning. In order to improve the accuracy and reliability of ship positioning, the positioning method combines the Dead Reckoning (DR) algorithm and the Global Positioning System (GPS). Experimental results of an illustrative example show the superior performance of the new algorithms when applied to ship positioning.

A DDF-based IMM-TFS Approach for the Accuracy Evaluation Problem of Rapid Transfer Alignment

This study aims to address the accuracy evaluation problem for rapid transfer alignment with the coexistence of large misalignment angles and uncertain observation noises. For the requirement of accuracy evaluation, complete information in terms of misalignment angles should be estimated during the alignment process. Thus, a fixed-interval smoothing approach is the core of solving this problem. In this paper, a new Divided Difference Filter (DDF)-based an Interacting Multiple Model Two-Filter Smoother (IMM-TFS) is developed to estimate the misalignment angles. The proposed DDF-based IMM-TFS releases the restriction of inverse nonlinearity by using the weighted statistical linearization regression method, and the resulting pseudo-linear model can be used for backward-time IMM filtering. The smoothing step takes into account the merging of estimations and the interaction of multiple models simultaneously. The new smoother is compared with the previous well-known methodologies in simulations. The results show that the DDF-based IMM-TFS can achieve better accuracy for misalignment angles estimation, and has a high efficiency for detecting the changes in a model.

A Maximum Correntropy Divided Difference Filter for Cooperative Localization

This paper derives a new maximum correntropy divided difference filter (DDF) to address the heavy-tailed measurement noise induced by non-Gaussian measurements in cooperative localization of autonomous underwater vehicles. By integrating the advantages of both the DDF and the maximum correntropy criterion, the proposed filter exhibits localization accuracy and robustness to address the heavy-tailed impulsive noise. The proposed maximum correntropy DDF has been tested through a lake trial. Experimental results indicate the superior performance of the proposed algorithm.

Robust detection of false data injection attacks for data aggregation in an Internet of Things-based environmental surveillance

Data aggregation is an important technology for environmental surveillance based on the Internet of Things (IoT) in the process of compressing redundant data collected from small devices that are distributed over a wide area in a network. However, since most IoT devices work in an unattended manner with limited security guarantees, they are extremely vulnerable to node compromise. Once adversaries take charge of compromised nodes, they can launch false data injection (FDI) attacks, which are known to be destructive for data aggregation. To minimize the damage caused by an FDI attack, we adopt a Hierarchical Bayesian Spatial-Temporal (HBST) model to describe the statistical characteristics of sensory data in an aggregation-based communication mode and propose an anomaly detection-based scheme to detect compromised nodes in the early stage. The basic idea behind our scheme is using state estimation techniques based on divided difference filtering (DDF) to detect false aggregated data and determining the nodes that are suspected of injecting false data using sequential hypothesis testing (SHT). Additionally, we model the problem of FDI attack detection using a quantitative two-player game theory analysis, derive the optimal strategies for both the adversaries and defenders, and demonstrate that the adversaries’ gain from the attack is greatly limited by the defenders, even in the worst case when both players follow their respective optimal strategies. Moreover, we present theoretical and simulation analyses to evaluate the performance of the proposed scheme in terms of the effectiveness, efficiency and overhead. The analytical results show that the proposed scheme achieves a high detection rate and low false positive rate with a small amount of detection samples.

Experimental implementation of MIMO model predictive controller-based second order divided difference filter for nonlinear systems

In this paper, we propose an experimental implementation of a nonlinear predictive controller based on the second order divided difference filter (DDF2) which is a derivate-free estimator. The used state estimator avoids the determination of Jacobian matrices required with the extended Kalman filter (EKF) for an easy implementation with nonlinear systems. In addition, the second order Stirling's interpolation of the estimator allows a more accurate a priori state prediction. A constrained nonlinear predictive controller-based DDF2 is designed to control a multivariable three-tank system. In order to generate the control signals, the controller solves a nonlinear objective function. To overcome the offset output error that appeared due to the inaccuracy of the system model, an internal state model correction scheme is adopted at each iteration. Practical results show the reliability of the proposed method in state estimation, setpoint tracking, and smooth control signal generation. The proposed controller outperforms the model predictive control based on linear state space model derived from the linearisation of the system model around the same setpoint trajectory.

EM‐based adaptive divided difference filter for nonlinear system with multiplicative parameter

SummaryThis paper is concerned with application of expectation maximization (EM) algorithm for deriving an adaptive version of divided difference filter for joint state estimation and multiplicative parameter identification of nonlinear system with the colored measurement noise. Owing to the fact that there exist a mutual coupling and interaction of state and parameter on each other, it requires a joint or simultaneous estimation of both state and parameter by a mutual iteration, and justly, EM iterates Expectation (E‐)step and Maximization (M‐)step to meet such requirement. Firstly, E‐step involves state filtering and smoothing issues under knowing the previous parameter identification results, which is well solved by resorting to the Gaussian approximation with a trade‐off between accuracy and complexity. Further, such Gaussian approximation estimators are applied for evaluating the condition expectation of complete‐data likelihood function, nonlinearly characterized by the multiplicative parameter needed to be optimized. Secondly, M‐step deals with the maximization of the condition expectation by directly making its derivative as zero to obtain the current general parameter identification equation as the nonlinear integral. Thirdly, by iteratively operating E‐step and M‐step, an adaptive divided difference filter is proposed for joint state estimation and parameter identification by using the second‐order Stirling interpolation to compute the associated nonlinear integral. Finally, the robust performance of the EM‐based adaptive version of divided difference filter to the unknown or time‐varying multiplicative parameter, as compared with the standard augmentation method, is demonstrated by a maneuvering target tracking example. Copyright © 2016 John Wiley & Sons, Ltd.

Augmentation method of XPNAV in Mars orbit based on Phobos and Deimos observations

Autonomous navigation for Mars probe spacecraft is required to reduce the operation costs and enhance the navigation performance in the future. X-ray pulsar-based navigation (XPNAV) is a potential candidate to meet this requirement. This paper addresses the use of the Mars’ natural satellites to improve XPNAV for Mars probe spacecraft. Two observation variables of the field angle and natural satellites’ direction vectors of Mars are added into the XPNAV positioning system. The measurement model of field angle and direction vectors is formulated by processing satellite image of Mars obtained from optical camera. This measurement model is integrated into the spacecraft orbit dynamics to build the filter model. In order to estimate position and velocity error of the spacecraft and reduce the impact of the system noise on navigation precision, an adaptive divided difference filter (ADDF) is applied. Numerical simulation results demonstrate that the performance of ADDF is better than Unscented Kalman Filter (UKF) DDF and EKF. In view of the invisibility of Mars’ natural satellites in some cases, a visibility condition analysis is given and the augmented XPNAV in a different visibility condition is numerically simulated. The simulation results show that the navigation precision is evidently improved by using the augmented XPNAV based on the field angle and natural satellites’ direction vectors of Mars in a comparison with the conventional XPNAV.

Modeling and robust controlling of laser welding process on high strength titanium alloy using fuzzy basis function networks and robust Takagi-Sugeno fuzzy controller

The purpose of this paper is to present and implement a novel approach to data-driven modeling and robust control of a laser keyhole welding process. The objective is to maintain the penetration depth of the laser welding process at a desired value under system uncertainties. A framework was proposed to estimate the keyhole diameter and the keyhole penetration depth in real time by type-1 and type-2 fuzzy basis function networks, and an adaptive divided difference filter. A robust Takagi-Sugeno fuzzy controller was implemented to adjust the laser power of the welding process. Experimental results demonstrated that the fuzzy models provided an accurate estimation of both the welding geometry and its variations due to uncertainties, and the robust Takagi-Sugeno fuzzy controller successfully reduced the penetration depth variation and improved the quality of the welding process.

Nonlinear regression Huber-based divided difference filtering

This article derives a nonlinear regression Huber-based divided difference filtering algorithm using a nonlinear regression approach for dynamic state estimation problems with non-Gaussian noises and outliers. In this approach, the nonlinear measurement model is directly used without linear or statistically linear approximation and the Huber-based divided difference filtering problem is solved using a Gauss–Newton approach. This new proposed filter method is then applied to a benchmark problem of estimating the trajectory of an entry body from discrete-time range data measured by a radar tracking station. Simulation results demonstrate the superior performance of the proposed filter as compared to the previous filter algorithms in the presence of non-Gaussian uncertainties.

Nonlinear Parameters and State Estimation for Adaptive Nonlinear Model Predictive Control Design

This paper deals with an adaptive nonlinear model predictive control (NMPC) based estimator in cases of mismatch modeling, presence of perturbations and/or parameter variations. Thus, we propose an adaptive nonlinear predictive controller based on the second-order divided difference filter (DDF) for multivariable systems. The controller uses a nonlinear state-space model for parameters and state estimation and for the control law synthesis. Two nonlinear optimization layers are included in the proposed algorithm. The first optimization problem is based on the output error (OE) model with a tuning factor, and it is dedicated to minimize the error between the model and the system at each sample time by estimating unknown parameters when assuming that all system states are available. The second optimization layer is used by the centralized nonlinear predictive controller to generate the control law which minimizes the error between future setpoints and future outputs along the prediction horizon. The proposed algorithm leads to a good tracking performance with an offset-free output and an effectiveness in perturbation attenuation. Practical results on a real setup show the reliability of the proposed approach.