Adaptive Divided Difference Filter Research Articles

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.

Orbit Determination Using Pulsar Timing Data and Orientation Vector

X-ray Pulsar Navigation (XPNAV) uses the Time Difference of Arrival (TDOA) of the pulsar signal between the spacecraft and Solar System Barycentre (SSB) to determine position. In this paper, a novel method to improve the performance of XPNAV via exploiting the pulsar position vector is proposed. First, the field of view of the collimator is utilised to find the pulsar orientation direction. Then, a searching strategy based on the modified Powell method under given coordinate frames is proposed. We also mathematically prove the existence of the extreme value of the searching strategy. Subsequently, an observation model based on the pulsar radiation vector is presented and applied to formulate the observation function together with pulsar time transfer function. Finally, an Adaptive Divided Difference Filter (ADDF) algorithm is introduced to iteratively estimate the position and velocity of the spacecraft. Numerical simulations show that the vector searching method is feasible and the pulsar radiation direction can improve the navigation performance by 75%. The simulation results also show that the ADDF performs better than Unscented Kalman Filtering (UKF) and DDF in position estimation.

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.

Adaptive divided difference filter for parameter and state estimation of non‐linear systems

An adaptive divided difference filter for joint estimation of parameters and states of a non-linear signal model has been proposed. The adaptive non-linear estimator, developed on the framework of second-order divided difference filter is intended for situations where the measurement noise statistics is unknown. Unlike other alternatives, the proposed non-linear adaptive estimator always ensures positive definiteness of the adapted measurement noise covariance. Performance of the evolved filter has been assessed with a bench mark non-linear problem of joint estimation of parameters and states. Simulation with Monte Carlo results demonstrate that the root-mean-square errors of estimated states and parameters are (i) better than those obtained from non-adaptive filters with same initial values of measurement error covariance and (ii) consistent with the estimated error covariance. Furthermore, it is shown that even when the measurement noise covariance varies with time the adapted measurement noise covariance can track the time-varying truth value.

Support Vector Regression-Based Adaptive Divided Difference Filter for Nonlinear State Estimation Problems

We present a support vector regression-based adaptive divided difference filter (SVRADDF) algorithm for improving the low state estimation accuracy of nonlinear systems, which are typically affected by large initial estimation errors and imprecise prior knowledge of process and measurement noises. The derivative-free SVRADDF algorithm is significantly simpler to compute than other methods and is implemented using only functional evaluations. The SVRADDF algorithm involves the use of the theoretical and actual covariance of the innovation sequence. Support vector regression (SVR) is employed to generate the adaptive factor to tune the noise covariance at each sampling instant when the measurement update step executes, which improves the algorithm’s robustness. The performance of the proposed algorithm is evaluated by estimating states for (i) an underwater nonmaneuvering target bearing-only tracking system and (ii) maneuvering target bearing-only tracking in an air-traffic control system. The simulation results show that the proposed SVRADDF algorithm exhibits better performance when compared with a traditional DDF algorithm.

Observer-Based Adaptive Robust Control of Friction Stir Welding Axial Force

Friction stir welding (FSW) is a relatively new and promising joining process that is the subject of much current research. When welding with constant parameters, the axial force can vary significantly due to changes in workpiece temperature and other process variations. These variations produce welds with inconsistent microstructure and tensile strength. Control of the axial weld force is desirable to improve the weld quality. In this paper, an observer-based adaptive robust control (ARC) approach for the axial force of FSW is presented to overcome process disturbances and model errors stemming from the simplistic dynamic models suitable for control. Some correlation is shown between spindle power and axial force, allowing readily available power measurements to be used for feedback. A model of the axial force is developed as a combination of a nonlinear static gain and linear dynamics. An axial force controller is constructed using the ARC approach and estimated state feedback from the adaptive divided difference filter (ADDF). Verification experiments are conducted on a vertical milling machine configured for FSW using an open architecture controller. The combined ARC/ADDF technique is shown to dramatically reduce axial force variations in the presence of significant process disturbances.