Low-cost adaptive square-root cubature Kalman filter forsystems with process model uncertainty

A novel low-cost adaptive square-root cubature Kalmanfilter （LCASCKF） is proposed to enhance the robustness of processmodels while only increasing the computational load slightly.It is well-known that the Kalman filter cannot handle uncertainties ina process model, such as initial state estimation errors, parametermismatch and abrupt state changes. These uncertainties severelyaffect filter performance and may even provoke divergence. Astrong tracking filter （STF）, which utilizes a suboptimal fading factor,is an adaptive approach that is commonly adopted to solvethis problem. However, if the strong tracking SCKF （STSCKF）uses the same method as the extended Kalman filter （EKF） tointroduce the suboptimal fading factor, it greatly increases thecomputational load. To avoid this problem, a low-cost introductorymethod is proposed and a hypothesis testing theory is applied todetect uncertainties. The computational load analysis is performedby counting the total number of floating-point operations and it isfound that the computational load of LCASCKF is close to that ofSCKF. Experimental results prove that the LCASCKF performs aswell as STSCKF, while the increase in computational load is muchlower than STSCKF.

Improved Square-Root UKF Algorithm for State Estimation of Nonlinear Systems

The square-root unscented Kalman filter （SR- UKF） for state estimation probably encounters the problem that Cholesky factor update of the covariance matrices can＇t be implemented when the zero＇th weight of sigma points is negative or the mnnerical computation error becomes large during the faltering procedure. Consequently the filter becomes invalid. An improved SR-UKF algorithm （ISR- UKF） is presented for state estimation of arbitrary nonlinear systems with linear measurements. It adopts a modified form of predicted covariance matrices, and modifies the Cholesky factor calculation of the updated covariance matrix originating from the square-root covariance filtering method. Discussions have been given on how to avoid the filter invalidation and further error accumulation. The comparison between the ISR-UKF and the SR-UKF by simulation also shows both have the same accuracy for state estimation. Finally the performance of the improved filter is evaluated under the impact of model mismatch. The error behavior shows that the ISR-UKF can overcome the impact of model mismatch to a certain extent and has excellent trace capability.

Underwater square-root cubature attitude estimator by use of quaternion-vector switching and geomagnetic field tensor

This paper presents a kind of attitude estimation algorithm based on quaternion-vector switching and square-root cubature Kalman filter for autonomous underwater vehicle(AUV).The filter formulation is based on geomagnetic field tensor measurement dependent on the attitude and a gyro-based model for attitude propagation. In this algorithm, switching between the quaternion and the three-component vector is done by a couple of the mathematical transformations. Quaternion is chosen as the state variable of attitude in the kinematics equation to time update, while the mean value and covariance of the quaternion are computed by the three-component vector to avoid the normalization constraint of quaternion. The square-root forms enjoy a continuous and improved numerical stability because all the resulting covariance matrices are guaranteed to stay positively semidefinite. The entire square-root cubature attitude estimation algorithm with quaternion-vector switching for the nonlinear equality constraint of quaternion is given. The numerical simulation of simultaneous swing motions in the three directions is performed to compare with the three kinds of filters and the results indicate that the proposed filter provides lower attitude estimation errors than the other two kinds of filters and a good convergence rate.

Sensor Fusion with Square-Root Cubature Information Filtering

This paper derives a square-root information-type filtering algorithm for nonlinear multi-sensor fusion problems using the cubature Kalman filter theory. The resulting filter is called the square-root cubature Information filter (SCIF). The SCIF propagates the square-root information matrices derived from numerically stable matrix operations and is therefore numerically robust. The SCIF is applied to a highly maneuvering target tracking problem in a distributed sensor network with feedback. The SCIF’s performance is finally compared with the regular cubature information filter and the traditional extended information filter. The results, presented herein, indicate that the SCIF is the most reliable of all three filters and yields a more accurate estimate than the extended information filter.

Multi-sensor Hybrid Fusion Algorithm Based on Adaptive Square-root Cubature Kalman Filter

In the normal operation condition, a conventional square-root cubature Kalman filter （SRCKF） gives sufficiently good estimation results. However, if the measurements are not reliable, the SRCKF may give inaccurate results and diverges by time. This study introduces an adaptive SRCKF algorithm with the filter gain correction for the case of measurement malfunctions. By proposing a switching criterion, an optimal filter is selected from the adaptive and conventional SRCKF according to the measurement quality. A subsystem soft fault detection algorithm is built with the filter residual. Utilizing a clear subsystem fault coefficient, the faulty subsystem is isolated as a result of the system reconstruction. In order to improve the performance of the multi-sensor system, a hybrid fusion algorithm is presented based on the adaptive SRCKF. The state and error covariance matrix are also predicted by the priori fusion estimates, and are updated by the predicted and estimated information of subsystems. The proposed algorithms were applied to the vessel dynamic positioning system simulation. They were compared with normal SRCKF and local estimation weighted fusion algorithm. The simulation results show that the presented adaptive SRCKF improves the robustness of subsystem filtering, and the hybrid fusion algorithm has the better performance. The simulation verifies the effectiveness of the proposed algorithms.

Square-root divided difference Rauch-Tung-Striebel smoother

A square-root version of the divided difference Rauch-Tung-Striebel (RTS) smoother is proposed in this paper. The square-root variant essentially propagates the square roots of the covariance matrices and can consistently improve the numerical stability because all the resulting covariance matrices are guaranteed to stay positive semi-definite. Furthermore, the square-root form ensures reliable implementation in an embedded system with fixed or limited precision although it is algebraically equivalent to the standard form. The new smoothing algorithm is tested in a challenging two-dimensional maneuvering target tracking problem with unknown and time-varying turn rate, and its performance is compared with that of other de-facto standard filters and smoothers. The simulation results indicate that the proposed RTS smoother markedly outperforms the associated filters and gives slightly smaller error than an unscented-based RTS smoother.

Robust SLAM using square-root cubature Kalman filter and Huber's GM-estimator

Mobile robot systems performing simultaneous localization and mapping(SLAM) are generally plagued by non-Gaussian noise.To improve both accuracy and robustness under non-Gaussian measurement noise,a robust SLAM algorithm is proposed.It is based on the square-root cubature Kalman filter equipped with a Huber' s generalized maximum likelihood estimator(GM-estimator).In particular,the square-root cubature rule is applied to propagate the robot state vector and covariance matrix in the time update,the measurement update and the new landmark initialization stages of the SLAM.Moreover,gain weight matrices with respect to the measurement residuals are calculated by utilizing Huber' s technique in the measurement update step.The measurement outliers are suppressed by lower Kalman gains as merging into the system.The proposed algorithm can achieve better performance under the condition of non-Gaussian measurement noise in comparison with benchmark algorithms.The simulation results demonstrate the advantages of the proposed SLAM algorithm.

Robust fault diagnosis for non-Gaussian stochastic systems based on the rational square-root approximation model

The task of robust fault detection and diagnosis of stochastic distribution control （SDC） systems with uncertainties is to use the measured input and the system output PDFs to still obtain possible faults information of the system. Using the rational square-root B-spline model to represent the dynamics between the output PDF and the input, in this paper, a robust nonlinear adaptive observer-based fault diagnosis algorithm is presented to diagnose the fault in the dynamic part of such systems with model uncertainties. When certain conditions are satisfied, the weight vector of the rational square-root B-spline model proves to be bounded. Conver- gency analysis is performed for the error dynamic system raised from robust fault detection and fault diagnosis phase. Computer simulations are given to demon- strate the effectiveness of the proposed algorithm.

HBFNEnKF混合同化方法设计及检验

基于前后张驰逼近(Back and Forth Nudging,简称BFN)和集合卡尔曼滤波(En KF)方法,构建了一种新的同化方法 HBFNEn KF(Hybrid Back and Forth Nudging En KF)混合同化方法,并将此同化系统分别与通道浅水模式(shallow water model)和全球浅水模式对接,检验了HBFNEn KF同化方法的有效性。同时,对比了集合均方根滤波(En SRF)、HNEn KF(Hybrid Nudging En KF)、HBFNEn KF三种方法在有误差模式中的同化效果。试验结果表明:HBFNEn KF同化方法保留了HNEn KF方法的同化连续性,解决了En KF同化不连续不平滑的问题,同时还有着更快的收敛速度;当采用单变量分析试验时,HBFNEn KF方法的优势最为明显,表明HBFNEn KF能够较好地保持不同模式变量间的平衡。此外,增量场尺度分析结果表明:相比En SRF,HBFNEn KF在大尺度范围有更好的同化效果,同时能够避免在中小尺度范围内出现大量的虚假增量。

Major Development Under Gaussian Filtering Since Unscented Kalman Filter

Filtering is a recursive estimation of hidden states of a dynamic system from noisy measurements.Such problems appear in several branches of science and technology,ranging from target tracking to biomedical monitoring.A commonly practiced approach of filtering with nonlinear systems is Gaussian filtering.The early Gaussian filters used a derivative-based implementation,and suffered from several drawbacks,such as the smoothness requirements of system models and poor stability.A derivative-free numerical approximation-based Gaussian filter,named the unscented Kalman filter(UKF),was introduced in the nineties,which offered several advantages over the derivativebased Gaussian filters.Since the proposition of UKF,derivativefree Gaussian filtering has been a highly active research area.This paper reviews significant developments made under Gaussian filtering since the proposition of UKF.The review is particularly focused on three categories of developments:i)advancing the numerical approximation methods;ii)modifying the conventional Gaussian approach to further improve the filtering performance;and iii)constrained filtering to address the problem of discrete-time formulation of process dynamics.This review highlights the computational aspect of recent developments in all three categories.The performance of various filters are analyzed by simulating them with real-life target tracking problems.

Information Protection Based on Extraction of Square Roots of Gaussian Integers

A cryptosystem based on computation of square roots of complex integers modulo composite n is described in this paper. This paper provides an algorithm extracting a square root of Gaussian integer. Various properties of square roots and a method for finding Gaussian generators are demonstrated. The generators can be instrumental in constructing other cryptosystems. It is shown how to significantly reduce average complexity of decryption per each block of ciphertext.

SCKF-STF-CN:a universal nonlinear filter for maneuver target tracking

Square-root cubature Kalman filter (SCKF) is more effective for nonlinear state estimation than an unscented Kalman filter.In this paper,we study the design of nonlinear filters based on SCKF for the system with one step noise correlation and abrupt state change.First,we give the SCKF that deals with the one step correlation between process and measurement noises,SCKF-CN in short.Second,we introduce the idea of a strong tracking filter to construct the adaptive square-root factor of the prediction error covariance with a fading factor,which makes SCKF-CN obtain outstanding tracking performance to the system with target maneuver or abrupt state change.Accordingly,the tracking performance of SCKF is greatly improved.A universal nonlinear estimator is proposed,which can not only deal with the conventional nonlinear filter problem with high dimensionality and correlated noises,but also achieve an excellent strong tracking performance towards the abrupt change of target state.Three simulation examples with a bearings-only tracking system are illustrated to verify the efficiency of the proposed algorithms.

MANY-SERVER QUEUES WITH CUSTOMER ABANDONMENT:A SURVEY OF DIFFUSION AND FLUID APPROXIMATIONS

The performance of a call center is sensitive to customer abandonment. In this survey paper, we focus on G/GI/n ＋ GI parallel-server queues that serve as a building block to model call center operations. Such a queue has a general arrival process （the G ）, independent and identically distributed （iid） service times with a general distribution （the first G1 ）, and iid patience times with a general distribution （the ＋GI）. Following the square-root safety staffing rule, this queue can be operated in the quality- and efficiency-driven （QED） regime, which is characterized by large customer volume, the waiting times being a fraction of the service times, only a small fraction of customers abandoning the system, and high server utilization. Operational efficiency is the central target in a system whose staffing costs dominate other expenses. If a moderate fraction of customer abandonment is allowed, such a system should be operated in an overloaded regime known as the efficiency-driven （ED） regime. We survey recent results on the many-server queues that are operated in the QED and ED regimes. These results include the performance insensitivity to patience time distributions and diffusion and fluid approximate models as practical tools for performance analysis.

Discussion on several statistical problems in establishing quality standards of standard decoctions

Since 2016,a number of studies have been published on standard decoctions used in Chinese medicine.However,there is little research on statistical issues related to establishing the quality standards for standard decoctions.In view of the currently established quality standard methods for standard decoctions,an improvement scheme is proposed from a statistical perspective.This review explores the requirements for dry matter yield rate data and index component transfer data for the application of two methods specified in‘‘Technical Requirements for Quality Control and Standard Establishment of Chinese Medicine Formula Granules,"which include the average value plus or minus three times the standard deviation (■±3SD) or 70%to 130%of the average value (■±30%■).The square-root arcsine transformation method is used as an approach to solve the problem of unreasonable standard ranges of standard decoctions.This review also proposes the use of merged data to establish a standard.A method to judge whether multiple sets of standard decoction data can be merged is also provided.When multiple sets of data have a similar central tendency and a similar discrete tendency,they can be merged to establish a more reliable quality standard.Assuming that the dry matter yield rate and transfer rate conform to a binomial distribution,the number of batches of prepared slices that are needed to establish the standard decoction quality standard is estimated.It is recommended that no less than 30 batches of prepared slices should be used for the establishment of standard decoction quality standards.