广义容积卡尔曼滤波

传统CKF采用三阶球面径向容积定律来计算非线性积分,该定律将球面数值积分与径向积分相结合,难以构造高阶CKF算法。此外,CKF在许多非线性问题上表现出估计精度低等问题。为了解决以上问题,提出了一种广义CKF族,所提算法彻底抛弃了球面径向积分定律。进一步指出,传统CKF是这种滤波算法的特殊形式。实验结果表明,高阶CKF比传统的非线性滤波器准确性更高。     

均方根嵌入式容积卡尔曼滤波

传统容积卡尔曼滤波(CKF)的基础是三阶球面-径向容积准则,该准则不仅要求计算n维超球体上的面积分,还需将容积准则与扩展高斯-拉盖尔准则配合使用,不易推导出高阶CKF滤波算法.此外,CKF推导所采用的三阶球面容积准则也存在缺陷,这极大地限制了CKF的滤波精度.为避免以上问题,本文基于嵌入式容积准则和均方根滤波技术,提出一种加性噪声环境下,用于非线性动态系统状态估计的全新容积卡尔曼滤波算法-三阶均方根嵌入式容积卡尔曼滤波(SICKF).SICKF具有滤波精度高、数值稳定性强等诸多优点,适用于动态目标跟踪、非线性系统控制等.仿真结果表明,SICKF的滤波精度显著优于传统的非线性滤波算法.     

A New Derivation of the Cubature Kalman Filters

This paper investigates the cubature Kalman filtering (CKF) for nonlinear dynamic systems. This third‐degree rule based filter employs a spherical‐radial cubature rule to numerically compute the integrals encountered in nonlinear filtering problems, thereby removing the requirements of explicitly computing the Jacobians. The cubature rule, however, requires computing the intractable integrals over a high‐dimensional spherical region for multidimensional applications. Moreover, the cubature formula that has been used to construct the spherical cubature formula has some demerits, most notably its inconvenient properties in computation and low estimation accuracy. Aimed at these issues, a general class of CKFs that uses only cubature rules is derived in this paper. It can be shown that the conventional CKF is a special case of the proposed algorithm. The paper also includes higher‐degree CKFs, especially two representative types of the fifth‐degree CKFs. Performance of the proposed algorithms is demonstrated via two target tracking problems. The experimental results, presented herein, illustrate the superior performance of higher‐degree CKFs to conventional nonlinear filters.     

基于cubature Kalman filter的INS/GPS组合导航滤波算法

INS/GPS组合导航系统的本质是非线性的,为改善非线性下INS/GPS组合导航精度,提出将一种新的非线性滤波cubature Kalman filter（CKF）应用于INS/GPS组合导航中.为此,建立了基于平台失准角的非线性状态模型和以速度误差及位置误差描述的观测模型,分析了CKF滤波原理,设计了INS/GPS组合滤波器,对组合导航非线性模型进行了仿真.仿真结果显示,相对于扩展卡尔曼滤波（EKF）,CKF降低了姿态、位置和速度估计误差,CKF更适合于处理组合导航的状态估计问题.     

Distributed fusion cubature Kalman filters for nonlinear systems

This paper is concerned with the distributed fusion estimation problem for multisensor nonlinear systems. Based on the Kalman filtering framework and the spherical cubature rule, a general method for calculating the cross‐covariance matrices between any two local estimators is presented for multisensor nonlinear systems. In the linear unbiased minimum variance sense, based on the cross‐covariance matrices, a distributed fusion cubature Kalman filter weighted by matrices (MW‐CKF) is presented. The proposed MW‐CKF has better accuracy and robustness. An example verifies the effectiveness of the proposed algorithms.     

Stochastic stability of cubature predictive filterCubature 预测滤波的随机稳定性分析

In this paper, the cubature predictive filter (CPF) is derived based on a third-degree spherical-radial cubature rule. It provides a set of cubature-points scaling linearly with the state-vector dimension, which makes it possible to numerically compute multivariate moment integrals encountered in the nonlinear predictive filter (PF). In order to facilitate the new method, the algorithm CPF is given firstly. Then, the theoretical analyses demonstrate that the estimated accuracy of the model error and system for the proposed CPF is higher than that of the traditional PF. Moreover, the authors analyze the stochastic boundedness and the error behavior of CPF for general nonlinear systems in a stochastic framework. In particular, the theoretical results present that the estimation error remains bounded and the covariance keeps stable if the system’s initial estimation error, disturbing noise terms as well as the model error are small enough, which is the core part of the CPF theory. All of the results have been demonstrated by numerical simulations for a nonlinear example system.     

一种带多步随机延迟量测高斯滤波器的一般框架解

提出了一种适用于线性和非线性系统的带多步随机延迟量测高斯滤波器的一般框架解. 为了完成状态的递归更新估计, 噪声向量和先前时刻状态向量被扩展到当前时刻状态向量中. 然后基于贝叶斯方法推导了扩展后状态向量的一般框架解. 对于非线性系统, 通过利用不同的数值计算方法计算贝叶斯解中的高斯加权积分可以推导获得不同的高斯近似滤波器. 最后本文利用三阶球径容积准则来实施提出的方法, 并通过量测被随机延迟多步的目标跟踪模型对所提出的方法进行了仿真, 仿真结果验证了提出方法的有效性和优点.     

Cubature卡尔曼滤波-卡尔曼滤波算法

针对条件线性高斯状态空间模型,提出cubature卡尔曼滤波-卡尔曼滤波算法(CKF-KF),分别应用CKF和KF估计模型中的非线性和线性状态.该算法对非线性与线性状态均进行cubature采样,并将两种样本通过线性方程和量测方程进行传播,以获得非线性状态估计.机动目标跟踪仿真结果表明,CKF-KF的估计精度比Rao-Blackwellized粒子滤波器(RBPF)略低,但算法运行时间不到其1％;与无迹卡尔曼滤波器(UKF-KF)相比,估计精度相当,但算法运行时间降低了22％,有效地提高了实时性.     

Cubature Kalman smoothers

The cubature Kalman filter (CKF) is a relatively new addition to derivative-free approximate Bayesian filters built under the Gaussian assumption. This paper extends the CKF theory to address nonlinear smoothing problems; the resulting state estimator is named the fixed-interval cubature Kalman smoother (FI-CKS). Moreover, the FI-CKS is reformulated to propagate the square-root error covariances. Although algebraically equivalent to the FI-CKS, the square-root variant ensures reliable implementation when committed to embedded systems with fixed precision or when the inference problem itself is ill-conditioned. Finally, to validate the formulation, the square-root FI-CKS is applied to track a ballistic target on reentry.     

容积法则辅助的交互式多模型滤波算法

交互式多模型滤波(IMM) 的交互环节使得系统状态量不再服从单纯的高斯分布, 用现有方法对其概率分布的估计存在较大的误差. 对此, 考虑到模型的混合概率是时变的, IMM的交互过程可以用非线性方程来描述, 因而采用容积卡尔曼滤波(CKF) 中的容积法则对高斯随机变量经非线性函数传播后的概率分布进行估计, 并从理论上证明了容积法则的近似精度. 仿真实验表明, 由于提高了对交互后随机变量概率分布的估计精度, 所提出的方法能够有效改善IMM在量测噪声较大时的滤波效果.

     

平方根体积积分卡尔曼滤波

体积积分是一种新的具有较高代数精度的积分方法。为了提高非线性滤波算法的精度和数值稳定性,将体积积分规则和平方根分解引入卡尔曼滤波框架中,提出了平方根体积积分卡尔曼滤波算法（SRCQKF）。新算法采用球半径体积规则和高斯-拉盖尔积分规则计算积分点,利用矩阵的QR分解得到协方差矩阵的平方根并传播平方根。两个典型的非线性系统的实验结果表明,与体积卡尔曼滤波相比,新算法提高了非线性状态的估计精度,具有较高的数值稳定性。     

基于交互容积卡尔曼的随机切换系统故障诊断

针对一类随机切换非线性系统的故障检测和故障估计问题,提出了一种基于交互式多模型和容积卡尔曼滤波（IMM CKF）的系统状态估计算法。该算法利用容积卡尔曼滤波（CKF）在不同时刻对每个子系统进行状态估计,把不同子系统状态估计结果融合得到最终的状态估计,实现对系统真实状态的估计。针对一类随机切换非线性系统发生执行器故障,采用IMM CKF估计系统状态;然后分析了IMM CKF算法的稳定性;根据状态估计结果,构造残差信号,设计残差评价函数,检测故障发生。当检测到故障发生时,设计增广系统,对故障幅值进行估计。通过仿真实验验证提出算法的有效性,结果表明该算法可以较为准确地诊断系统故障。     

High-degree cubature Kalman filter

The cubature Kalman filter (CKF), which is based on the third degree spherical–radial cubature rule, is numerically more stable than the unscented Kalman filter (UKF) but less accurate than the Gauss–Hermite quadrature filter (GHQF). To improve the performance of the CKF, a new class of CKFs with arbitrary degrees of accuracy in computing the spherical and radial integrals is proposed. The third-degree CKF is a special case of the class. The high-degree CKFs of the class can achieve the accuracy and stability performances close to those of the GHQF but at lower computational cost. A numerical integration problem and a target tracking problem are utilized to demonstrate the necessity of using the high-degree cubature rules to improve the performance. The target tracking simulation shows that the fifth-degree CKF can achieve higher accuracy than the extended Kalman filter, the UKF, the third-degree CKF, and the particle filter, and is computationally much more efficient than the GHQF.     

A cubature H∞ filter and its square-root version

In this paper, we present a nonlinear state estimation algorithm based on the fusion of an extended H_∞ (EH_∞F) and a cubature Kalman filter (CKF); the resulting estimator is called a cubature H_∞ filter. The recently developed CKF is a Gaussian approximation of a Bayesian filter and its performance over non-Gaussian noises may degrade. In contrast, the H_∞ filter is capable of estimating the states of linear systems with non-Gaussian noises and the extended H_∞ filter (EH_∞F) can estimate the states of non-linear and non-Gaussian systems. Similar to the H_∞ filter, an EH_∞F also does not make any assumptions about the statistics of the process or measurement noise, but it does require Jacobians during the state estimation of nonlinear systems, which degrade the overall performance when the nonlinearities are severe. The cubature H_∞ filter is developed to have the desirable features of both CKF and EH_∞F. For numerical accuracy, a square-root version of the cubature H_∞ filter is developed using J-unitary transformation. The efficacy of the square-root cubature H_∞ filter is verified on continuous stirred tank reactor and permanent magnet synchronous motor examples.     

基于正交变换的改进CKF算法

为了解决容积卡尔曼滤波(CKF)算法在处理高维问题时出现的非局部采样问题,提出基于采样点正交变换的改进CKF算法(TCKF).从数值积分近似角度导出无迹卡尔曼滤波(UKF)和CKF两种近似滤波算法,并指出CKF只是UKF的一个特例;基于多元Taylor级数展开分析,揭示CKF在克服UKF数值不稳定性问题的同时,引入非局部采样问题;对Cubature点集进行正交变换得到TCKF算法,并从理论上证明,在高维、强非线性等非局部采样问题突出的滤波模型中,TCKF具有比CKF更高的估计精度.仿真实例验证了所提出算法的有效性.     

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.     

交互式容积卡尔曼滤波及其应用 交互式容积卡尔曼滤波及其应用

针对非线性系统模型参数未知情况下的状态估计问题, 提出一种融合极大后验估计的交互式容积卡尔曼滤波算法(InCKF). 该算法利用二阶斯特林插值公式和无迹变换对非线性函数的近似思想, 实现对模型未知参数的确定, 从而使滤波算法摆脱对模型参数精确已知的依赖, 并通过容积卡尔曼滤波算法完成状态估计和量测更新. 仿真结果表明, 相比于经典的参数扩维方法, InCKF 算法具有更高的精度和更强的数值稳定性.

     

改进型CKF算法及其在GNSS/INS中的应用

针对容积卡尔曼滤波算法(CKF)在高阶非线性系统应用中,局部容积点采样不准确及系统状态突变导致滤波精度降低甚至发散的问题,提出一种改进型CKF(TSCKF)算法.采用新的容积点采样规则提高非线性函数映射准确性,进而提高滤波精度;引入强跟踪滤波(STF)的渐消因子,提升算法在状态突变时的鲁棒性.将此算法应用于GNSS/INS(Global Navigation Satellite System/Inertial Navigation System)紧组合导航系统中,仿真结果表明,该算法能够显著提高导航精度,鲁棒性强,在状态突变情况下依然可以满足导航精度要求.     

基于均方根容积卡尔曼的δ-GLMB多目标跟踪算法

在非线性高杂波密度场景下,高斯混合(Gaussian Mixture,GM)实现的δ-广义标签多伯努利滤波器(δ-Generalized Labeled Multi-Bernoulli Filter,δ-GLMB)难以准确地估计目标数目及运动状态。针对这一问题,提出基于均方根容积卡尔曼滤波(Square-rooted Cubature Kalman Filter,SCKF)的δ-GLMB高斯混合实现算法。基于三阶球面-径向容积准则选取一组等权的容积点集,对GM-δ-GLMB滤波器的伯努利分量传递过程中的高斯参量进行预测及更新,实现非线性模型系统下的目标跟踪。仿真结果表明,与现有的δ-GLMB滤波器的扩展卡尔曼滤波(Extended Kalman Filter,EKF)高斯混合实现及无迹卡尔曼滤波(Unscented Kalman Filter,UKF)高斯混合实现相比,该算法可提高非线性高杂波密度环境下的目标跟踪精度。     

Cubature Kalman Filter Under Minimum Error Entropy With Fiducial Points for INS/GPS Integration

Traditional cubature Kalman filter(CKF)is a preferable tool for the inertial navigation system(INS)/global positioning system(GPS)integration under Gaussian noises.The CKF,however,may provide a significantly biased estimate when the INS/GPS system suffers from complex non-Gaussian disturbances.To address this issue,a robust nonlinear Kalman filter referred to as cubature Kalman filter under minimum error entropy with fiducial points(MEEF-CKF)is proposed.The MEEF-CKF behaves a strong robustness against complex nonGaussian noises by operating several major steps,i.e.,regression model construction,robust state estimation and free parameters optimization.More concretely,a regression model is constructed with the consideration of residual error caused by linearizing a nonlinear function at the first step.The MEEF-CKF is then developed by solving an optimization problem based on minimum error entropy with fiducial points(MEEF)under the framework of the regression model.In the MEEF-CKF,a novel optimization approach is provided for the purpose of determining free parameters adaptively.In addition,the computational complexity and convergence analyses of the MEEF-CKF are conducted for demonstrating the calculational burden and convergence characteristic.The enhanced robustness of the MEEF-CKF is demonstrated by Monte Carlo simulations on the application of a target tracking with INS/GPS integration under complex nonGaussian noises.