Continuous‐discrete filters for bearings‐only underwater target tracking problems

In this work, we develop a continuous‐discrete shifted Rayleigh filter (CD‐SRF) and a continuous‐discrete sparse‐grid Gauss‐Hermite filter (CD‐SGHF) for a real‐life passive underwater bearings‐only target tracking problem. The stochastic difference equation describing the process model is derived from its continuous equivalent using Ito‐Taylor expansion of order 1.5. The performance of the proposed filters is compared in terms of root mean square error (RMSE), track divergence and computational time. For a fair comparison, popular filters like the unscented Kalman filter (UKF), cubature Kalman filter (CKF) and Gauss–Hermite filter (GHF) are implemented. The effect of initial uncertainty, measurement noise covariance and sampling time on filtering accuracy is also studied. Finally, RMSEs of all the filters are evaluated in comparison with the Cramer–Rao lower bound (CRLB). From simulation results, it was observed that CD filters performed with higher accuracy than their discrete equivalents, with CD‐SRF proving to be the most accurate among all the filters.     

Numerical robustness of extended Kalman filtering based state estimation in ill‐conditioned continuous‐discrete nonlinear stochastic chemical systems

This paper presents a case study investigation of numerical robustness of extended Kalman filters used for estimation of stochastic chemical systems with ill‐conditioned measurements. Here, we consider both a batch reactor model and that of a continuously stirred tank reactor. Our purpose is to explore performance of extended Kalman filtering–based state estimators when the measurement model becomes increasingly ill conditioned. In this way, we determine numerically robust methods, which are suitable for accurate estimation of stochastic chemical systems in the presence of round‐off and other disturbances. We examine both conventional filters and their square‐root forms. All these algorithms are implemented by means of the conventional matrix inversion used in their measurement update steps and the Moore‐Penrose pseudoinversion as well. Furthermore, the square‐root filters under investigation are obtained in two ways, namely, by solving square‐root moment differential equations and by square rooting the filter itself. We show that only the square‐root filters grounded in the second approach (with use of stable orthogonal decompositions) are numerically robust and provide the excellent estimation accuracy within all our ill‐conditioned stochastic chemical system scenarios considered in this paper. In addition, the convectional filters and the square‐root variants based on solving moment equations are rather sensitive to round‐off and may be useful and accurate if the chemical system at hand is rather well conditioned.     

Square‐Root Cubature‐Quadrature Kalman Filter

In this paper, an on‐going work introducing square‐root extension of cubature‐quadrature based Kalman filter is reported. The proposed method is named square‐root cubature‐quadrature Kalman filter (SR‐CQKF). Unlike ordinary cubature‐quadrature Kalman filter (CQKF), the proposed method propagates and updates square‐root of the error covariance without performing Cholesky decomposition at each step. Moreover SR‐CQKF ensures positive semi‐definiteness of the state covariance matrix. With the help of two examples we show the superior performance of SR‐CQKF compared to EKF and square root cubature Kalman filter.     

New minimum sigma set for unscented filtering

In this work, a new minimum set of sigma points for unscented filtering is proposed along with its unscented Kalman filter in both square‐root and nonsquare‐root forms. Comparative with the other reduced sigma sets of the literature, the new sigma set is, in some cases, better defined or, in another case, a generalization. In numerical examples, the unscented transform of the new sigma set is compared with the unscented transforms of the other reduced sigma sets of the literature. In addition, the performance of the new unscented Kalman filter is studied in an aircraft target tracking scenario. Copyright © 2014 John Wiley & Sons, Ltd.     

Unscented Kalman filter for continuous‐time nonlinear fractional‐order systems with process and measurement noises

This study proposes the design of unscented Kalman filter for a continuous‐time nonlinear fractional‐order system involving the process noise and the measurement noise. The nonlinear fractional‐order system is discretized to get the difference equation. According to the unscented transformation, the design method of unscented Kalman filter for a continuous‐time nonlinear fractional‐order system is provided. Compared with the extended Kalman filter, the proposed method can obtain a more accurate estimation effect. For fractional‐order systems containing non‐differentiable nonlinear functions, the method proposed in this paper is still effective. The unknown parameters are also discussed by the augmented vector method to achieve the state estimation and parameter identification. Finally, two examples are offered to verify the effectiveness of the proposed unscented Kalman filter for nonlinear fractional‐order systems.     

SVD‐aided EKF attitude estimation with UD factorized measurement noise covariance

This paper describes singular value decomposition (SVD) aided extended Kalman filter (EKF) for nanosatellite's attitude estimation. The development of the filter kinematic/dynamic model, the measurement models of the sun sensors, and the magnetometers used to generate vector measurements are presented. Vector measurements are used in SVD for satellite attitude determination purposes. In the proposed method, EKF inputs come from SVD as the linear measurements of attitude angles and their error covariance. In this step, UD factorizes the attitude angles error covariance, forming the measurements in order to obtain the appropriate inputs for the filtering stage. Results are presented and analyzed in addition to discussion of the sub‐step, which is the UD factorization on the measurement covariance. The accuracy of the estimation results of the SVD‐aided EKF with and without UD factorization is compared for estimation performance.     

A New Approach To Optimal and Self‐Tuning Filtering,Smoothing and Prediction For Discrete‐Time Systems

A new approach to optimal and self‐tuning state estimation of linear discrete time‐invariant systems is presented, using projection theory and innovation analysis method in time domain. The optimal estimators are calculated by means of spectral factorization. The filter, predictor, and smoother are given in a unified form. Comparisons are made to the previously known techniques such as the Kalman filtering and the polynomial method initiated by Kucera. When the noise covariance matrices are not available, self‐tuning estimators are obtained through the identification of an ARMA innovation model. The self‐tuning estimator asymptotically converges to the optimal estimator.     

Variants of extended Kalman filtering approaches for Bayesian tracking

We provide a tutorial for a number of variants of the extended Kalman filter (EKF). In these methods, so called, sigma points are employed to tackle the nonlinearity of problems. The sigma points exactly represent the mean and the variance of the state distribution function in a dynamic state equation. The initially developed EKF variant, that is, unscented Kalman filter (UKF) (also called sigma point Kalman filter) shows enhanced performance compared with that of conventional EKF in the literature. Another variant, which is not well known, is central difference Kalman filter (CDKF) whose way to approximate the nonlinearity is based on the Sterling's polynomial interpolation formula instead of the Taylor series. Endeavor to reduce the computational load resulted in the development of square root versions of both UKF and CDKF, that is, square root unscented Kalman filter and square root central difference Kalman filter (SR‐CDKF). These SR‐versions are supposed to be numerically more stable than their original versions because the state covariance is guaranteed to be positive definite by avoiding the step of matrix decomposition. In this paper, we provide the step‐by‐step algorithms of above‐mentioned EKF variants with their pros and cons. We apply these filtering methods to a number of problems in various disciplines for performance assessment in terms of both mean squared error (MSE) and processing speed. Furthermore, we show how to optimize the filters in terms of MSE performance depending on diverse scenarios. According to simulation results, CDKF and SR‐CDKF show the best MSE performance in most scenarios; particularly, SR‐CDKF shows faster processing speed than that of CDKF. Therefore, we justify that SR‐CDKF is the most efficient and the best approach among the Kalman variants including the EKF for various nonlinear problems. The motivation of this paper targets at the contribution to the disseminative usage of the Kalman variants approaches, particularly, SR‐CDKF taking advantage of its estimating performance and high processing speed. Copyright © 2016 John Wiley & Sons, Ltd.     

自适应快速弱敏无迹Kalman滤波算法

针对现有弱敏无迹Kalman滤波需要代数求解增益矩阵耗时长和不能实时调节敏感性权重的问题,提出一种自适应快速弱敏无迹Kalman滤波算法.该算法在弱敏控制技术的基础上,重新定义弱敏无迹Kalman滤波的敏感性权重矩阵,将状态估计误差对不确定参数的敏感性加入滤波的代价函数,并通过最小化该代价函数得到滤波增益矩阵的解析解,...     

有色噪声下的平方根UKF在天文自主导航中的应用

针对由星敏感器和光学导航相机组成的卫星天文自主导航系统, 传统的平方根UKF不能很好地解决测量噪声为有色噪声情况下的非线性滤波问题, 导致导航系统的精度下降. 为此, 提出了一种有色噪声情况下的平方根UKF方法. 同时, 为了避免在数值计算的过程中, 由于舍入误差而破坏误差协方差矩阵的正定性和对称性, 在整个递推计算过程中, 借鉴平方根Kalman滤波理论, 采用协方差矩阵平方根进行递推计算, 改善滤波算法的稳定性, 协方差矩阵的平方根更新用cholesky分解和qr分解来计算. 将该方法应用于卫星天文自主导航系统中, 实验仿真结果表明, 相对于传统的平方根UKF算法, 所设计的平方根UKF算法能够很好地解决测量噪声为有色噪声情况下估计精度低问题.     

Suboptimal filter for continuous‐time linear systems with unknown parameters

The filtering problem for continuous‐time linear systems with unknown parameters is considered. A new suboptimal filter is herein proposed. It is based on the optimal mean‐square linear combination of the local Kalman filters. In contrast to the optimal weights, the suboptimal weights do not depend on current observations; thus, the proposed filter can easily be implemented in real‐time. Examples demonstrate high accuracy and efficiency of the suboptimal filter. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A Seventh‐Degree Cubature Kalman Filter

The fifth‐degree cubature Kalman filter (CKF) has been proved to be a kind of algorithm that has higher precision than the third‐degree CKF and unscented Kalman filter (UKF). In order to further improve the performance of CKF, the seventh‐degree CKF is proposed in this paper by expanding the spherical‐radial rule, and a new kind of deterministic sampling method is derived based on the seventh‐degree cubature rule. Through the comparison in target tracking simulation, the seventh‐degree CKF methods are shown to be able to enhance filtering precision compared to the fifth‐degree CKF, the third‐degree CKF and the UKF filter.     

Unscented and square‐root unscented Kalman filters for quaternionic systems

Proper construction of an unscented Kalman filter (UKF) for unit quaternionic systems is not straightforward due to the incompatibility between the algebraic properties of the unit quaternions and the common real vector space operations (additions and scalar multiplications) needed in the steps of a filter algorithm. This work studies, in detail, all UKFs and square‐root UKFs for quaternionic systems proposed in the literature. First, we classify the algorithms according to the preservation of the unity norm of the quaternion variables. Second, we propose two new algorithms: the quaternionic additive unscented Kalman filter (QuAdUKF) and a square‐root variant of it. The QuAdUKF encompasses all known UKFs for quaternionic systems of the literature preserving, in all steps, the norm of the unit quaternion variables. Besides, it can also yield new UKFs with this norm preservation property. The QuAdUKF's square‐root variant has better properties in comparison with all the square‐root UKFs for quaternionic systems of the literature. Numerical experiments for a spacecraft attitude estimation problem illustrate the theoretical results.     

Measurement updating using the U-D factorization

The discrete linear filtering problem is treated by factoring the filter error covariance matrix as P = UDU^T. Efficient and stable measurement updating recursions are developed for the unit upper triangular factor, U, and the diagonal factor, D. This paper treats only the parameter estimation problem; effects of mapping, inclusion of process noise and other aspects of filtering are treated in separate publications. The algorithm is simple and, except for the fact that square roots are not involved, can be likened to square root filtering. Like the square root filter our algorithm guarantees nonnegativity of the computed covariance matrix. As is the case with the Kalman filter, our algorithm is well suited for use in real time. Attributes of our factorization update include: efficient one point at a time processing that requires little more computation than does the optimal but numerically unstable conventional Kalman measurement update algorithm; stability that compares with the square root filter and the variable dimension flexibility that is enjoyed by the square root information filter. These properties are the subject of this paper.     

On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems

This paper considers the application of the unscented Kalman filter (UKF) to continuous-time filtering problems, where both the state and measurement processes are modeled as stochastic differential equations. The mean and covariance differential equations which result in the continuous-time limit of the UKF are derived. The continuous-discrete UKF is derived as a special case of the continuous-time filter, when the continuous-time prediction equations are combined with the update step of the discrete-time UKF. The filter equations are also transformed into sigma-point differential equations, which can be interpreted as matrix square root versions of the filter equations.     

带有色量测噪声的非线性系统 Unscented 卡尔曼滤波器

传统Unscented卡尔曼滤波器(Unscented Kalman filter, UKF)要求噪声必须为高斯白噪声, 无法解 决带有色噪声的非线性系统滤波问题. 为此, 本文提出了一种带有色量测噪声的UKF滤 波新算法. 首先,基于量测信息增广和最小方差估计, 推导出一类带有色量测噪声的非 线性离散系统状态的最优滤波框架, 接着采用Unscented变换(Unscented transformation, UT)来计算最优框架中的 非线性状态后验均值和协方差, 进而得到有色量测噪声下UKF滤波递推公式. 所设 计的UKF新方法能有效地解决传统UKF在量测噪声有色情况下非线性滤波失效的问题, 数 值仿真实例验证了其可行性和有效性.     

基于平方根UKF的SLAM算法

同步定位与地图构建（SLAM）是移动机器人实现真正自主的关键,无迹卡尔曼滤波（UKF）由于直接利用系统非线性模型而在SLAM问题中得到广泛的应用。基于平方根滤波可以确保协方差矩阵的非负定的思想,将平方根UKF应用到SLAM问题中,确保了SLAM算法的稳定性,并得到了较高的估计精度。仿真结果表明,该算法是有效的。     

强跟踪平方根容积卡尔曼滤波和 自回归模型融合的故障预测

为了解决非线性系统中不可测量参数的预测问题,提出一种带有次优渐消因子的强跟踪平方根容积卡尔曼滤波(STSCKF)和自回归(AR)模型相结合的故障预测方法.利用AR模型时间序列预测法预测未来时刻的测量值,将预测的测量值作为STSCKF的测量变量,从而将预测问题转化为滤波估计问题.STSCKF通过在预测误差方差阵的均方根中引入渐消因子调节滤波过程中的增益矩阵,克服了故障参数变化函数未知情况下普通SCKF跟踪故障参数缓慢甚至失效的局限性,使得STSCKF能较好地预测故障参数的发展趋势.连续搅拌反应釜(CSTR)仿真结果表明,STSCKF的预测精度高于普通SCKF和强跟踪无迹卡尔曼滤波(STUKF),验证了方法的有效性.     

A robust discrete‐time adaptive control approach for systems with almost periodic time‐varying parameters

A periodic adaptive control approach is proposed for a class of nonlinear discrete‐time systems with time‐varying parametric uncertainties which are almost periodic, and the only prior knowledge is the periodicity. The new adaptive controller updates the parameters and the control signal periodically in a pointwise manner over one entire period, in the sequel that achieves a bounded tracking convergence. The result is further extended to scenarios with unknown input gain, higher order dynamics, and tracking. Copyright © 2012 John Wiley & Sons, Ltd.     

基于卡尔曼滤波预测的无偏量测转换方法

针对在目标跟踪系统中通常使用量测转换方法将球面坐标系下的量测量转换到笛卡尔坐标系下,传统量测转换方法在互距离测量误差增大时跟踪性能有所下降的问题,提出一种基于卡尔曼滤波预测的无偏量测转换方法。采用无偏量测转换方法变换量测量,基于卡尔曼滤波预测值,结合无迹变换算法估计转换量测方差。仿真结果表明,所提出的转换量测卡尔曼滤波算法较现有方法具有更高的跟踪精度和可信度。