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。     

基于平方根求积卡尔曼滤波器的弹道目标跟踪

针对再入阶段的弹道目标跟踪问题,提出运用平方根求积卡尔曼滤波器(SRQKF)估计目标的状态.所提出的算法是求积卡尔曼滤波(QKF)算法的平方根实现.该算法传播了目标状态的均值和协方差的平方根,确保了协方差矩阵的对称性和半正定性,改进了数值精度和稳定性,但其计算复杂性稍有增加.仿真实验表明,所提出算法的估计精度优于QKF算法和扩展卡尔曼滤波(EKF)算法,是一种很有效的非线性滤波方法.     

Hyperbolic‐singular‐value‐decomposition‐based square‐root accurate continuous‐discrete extended‐unscented Kalman filters for estimating continuous‐time stochastic models with discrete measurements

This paper presents novel square‐root accurate continuous‐discrete extended‐unscented Kalman filtering (ACD‐EUKF) algorithms for treating continuous‐time stochastic systems with discrete measurements. The time updates in such methods are fulfilled as those in the extended Kalman filter whereas their measurement updates are copied from the unscented Kalman filter. All this allows accurate predictions of the state mean and covariance to be combined with accurate measurement updates. The main weakness of this technique is the need for the Cholesky decomposition of predicted covariances derived in time‐update steps. Such a factorization is highly sensitive to numerical integration and round‐off errors committed, which may result in losing the covariance's positivity and, hence, failing the Cholesky decomposition. The latter problem is usually solved in the form of square‐root filtering implementations, which propagate not the covariance matrix but its square root instead. Here, we devise square‐root ACD‐EUKF methods grounded in the singular value decomposition (SVD). The SVD rooted in orthogonal transforms is applicable to any ACD‐EUKF with nonnegative weights, whereas the remaining ones, which can enjoy negative weights as well, are treated by means of the hyperbolic SVD based on J‐orthogonal transforms. The filters constructed are presented in a concise algorithmic form, which is convenient for practical use. Their two particular versions grounded in the classical and cubature unscented Kalman filtering parameterizations are examined in severe conditions of tackling a radar tracking problem, where an aircraft executes a coordinated turn. These are also compared to their non‐square‐root predecessor and other methods within the target tracking scenario with ill‐conditioned measurements.     

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.     

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.     

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.     

基于迭代平方根容积粒子滤波目标跟踪算法

针对粒子滤波目标跟踪算法粒子退化及跟踪精度问题,提出了一种基于马尔可夫链-蒙特卡罗(MCMC,Markov Chain Monte Carlo)的迭代平方根容积粒子滤波(ISRCPF,iterated square root cubature Kalman particle filter)算法(ISRCPF-MCMC).在该滤波算法中,利用容积数值积分原则计算非线性随机函数的均值和方差,通过正交矩阵分解代替矩阵开方,在生成的粒子滤波建议分布中融入当前量测值,提高对系统后验概率的逼近程度.然后在此基础上融合MCMC抽样算法(MH,Metropolis Hasting)对所选建议分布进行优化,增加粒子多样性,以提高跟踪精度.仿真试验结果表明,ISRCPF-MCMC算法的估计误差与其他算法相比降低至0.403％.     

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.     

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.     

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.     

Square root kalman algorithms in econometrics

This paper describes a sequential square root method which is aimed at solving the numerical problems affecting the conventional Kalman filter. Simple square root algorithms are derived for the Kalman covariance and information filters and for the smoothing equations. A comparison with other square root methods is also provided.     

Adaptive Gaussian filters for nonlinear state estimation with one-step randomly delayed measurements

This paper proposes new algorithms of adaptive Gaussian filters for nonlinear state estimation with maximum one-step randomly delayed measurements. The unknown random delay is modeled as a Bernoulli random variable with the latency probability known a priori. However, a contingent situation has been considered in this work when the measurement noise statistics remain partially unknown. Due to unavailability of the complete knowledge of measurement noise statistics, the unknown measurement noise covariance matrix is estimated along with states following: (i) variational Bayesian approach, (ii) maximum likelihood estimation. The adaptation algorithms are mathematically derived following both of the above approaches. Subsequently, a general framework for adaptive Gaussian filter is presented with which variants of adaptive nonlinear filters can be formulated using different rules of numerical approximation for Gaussian integrals. This paper presents a few of such filters, viz., adaptive cubature Kalman filter, adaptive cubature quadrature Kalman filter with their higher degree variants, adaptive unscented Kalman filter, and adaptive Gauss–Hermite filter, and demonstrates the comparative performance analysis with the help of a nontrivial Bearing only tracking problem in simulation. Additionally, the paper carries out relative performance comparison between maximum likelihood estimation and variational Bayesian approaches for adaptation using Monte Carlo simulation. The proposed algorithms are also validated with the help of an off-line harmonics estimation problem with real data.     

基于矩阵对角化变换的高阶容积卡尔曼滤波

为了提高高阶容积卡尔曼滤波器(CKF)的滤波性能, 提出一种基于矩阵对角化变换的高阶CKF 算法. 该算法基于高阶容积准则, 利用矩阵对角化变换代替标准高阶CKF 中的Cholesky 分解, 使得协方差矩阵分解后的平方根矩阵保留了原有的特征空间信息, 状态统计量计算更加准确, 从而提高了滤波精度; 同时, 矩阵对角化变换不要求协方差矩阵正定, 增强了算法滤波稳定性. 仿真结果表明, 所提出的算法是可行而有效的, 明显改善了标准高阶CKF 的滤波效果.

     

有色量测噪声下的改进求容积卡尔曼滤波器

为解决标准求容积卡尔曼滤波器在有色量测噪声条件下滤波精度退化的问题,提出改进求容积卡尔曼滤波器及其平方根形式.首先利用一阶马尔科夫模型白化非线性离散随机系统中有色量测噪声,将有色量测噪声下非线性离散随机系统转化为白噪声下非线性时滞系统.然后根据所得非线性时滞系统推导其高斯域的贝叶斯滤波框架,最后基于3度Spherical-Radial规则将该滤波框架近似为改进的求容积卡尔曼滤波器和其平方根形式.机动目标跟踪仿真试验结果表明两种改进求容积卡尔曼滤波算法在标准白噪声条件下与标准求容积卡尔曼滤波算法的估计精度相同,而在有色量测噪声背景下滤波精度和鲁棒性更优.     

基于迭代容积卡尔曼滤波的神经网络训练算法

针对现有应用非线性滤波算法对神经网络进行训练时存在精度不足的问题,提出了一种基于迭代容积卡尔曼滤波的神经网络训练算法。首先,将前馈神经网络各个节点的连接权值和偏置作为状态向量,建立前馈神经网络的状态空间模型。其次,利用Spherical-Radial准则生成容积点,并依据Gauss-Newton迭代策略来优化量测更新过程中获取的状态估计值和状态估计误差协方差,通过容积卡尔曼滤波估计精度的改善,提升神经网络节点的连接权值和偏置的训练效果。理论分析和仿真实验结果验证了所提算法的可行性和有效性。     

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.

     

适用于事件触发的分布式随机目标跟踪方法

研究了一类基于RSSI（Received signal strength indication）测距的分布式移动目标跟踪问题,提出了一种适用于事件触发无线传感器网络（Wireless sensor networks,WSNs）的分布式随机目标跟踪方法.首先考虑移动机器人模型的不确定性,引入了带有随机参数的过程噪声协方差,应用改进平方根容积卡尔曼滤波（Square root cubature Kalman filter,SRCKF）得到局部估计;然后采用无模型CI（Covariance intersection）融合估计方法以降低随机过程噪声协方差带来的不利影响.该方法充分利用有模型和无模型方法的优势,实现系统模型和量测不理想情况下的分布式目标跟踪.基于E-puck机器人的目标跟踪实验表明,事件触发的工作模式可有效地减少能量消耗,带随机参数的滤波方法更适合于随机目标的跟踪.     

基于自适应SR-CKF的序贯式WSNs目标跟踪算法

针对无线传感器网络(WSNs)动态目标跟踪问题,即通过对传感器获取的动态系统状态进行估计,预测目标的位置.提出一种基于自适应平方根容积卡尔曼(SR-CKF)的序贯式WSNs动态目标跟踪算法.该算法在运算过程中直接传递目标状态均值和协方差矩阵的平方根因子,降低了计算的复杂度.将目标跟踪过程序贯式地分配到动态簇集的每一个节点上,减小了无线通信过程中碰撞和干扰现象的发生,降低了节点通信和计算负担.针对不良观测信息,基于新息协方差匹配原理,建立了自适应SR-CKF,提高了整个系统的鲁棒性.实验仿真结果表明,本文提出的基于自适应SR-CKF的序贯式WSNs目标跟踪算法有效的提高了跟踪的精度和稳定性并且减小了传感器节点间通信的能量损耗.     

强跟踪平方根UKFNN的铝电解槽工耗动态演化模型

铝电解过程具有多变量、强耦合、强干扰、参数时变等特征,故其模型开发是一个技术难点. 根据该过程的特点,本文提出强跟踪平方根无迹Kalman神经网络（Strong tracking square root unscented Kalman filter neural network,STR-UKFNN）,并用其建立铝电解槽工艺能耗的动态演化模型. 该方法利用误差协方差矩阵的平方根代替UKFNN算法中的协方差阵,避免误差协方差矩阵可能出现负定而导致滤波发散,并在UKFNN算法中引入渐消因子和弱化因子,实时调整滤波增益,提高模型收敛速度和其对突变状态的跟踪能力. 通过某铝厂170kA预焙槽的日报样本验证表明,该方法提高了能耗模型的精度和对电解槽突变状态的实时跟踪能力,有助于指导铝电解过程操作参数的优化.     

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。