A survey of numerical methods for nonlinear filtering problems

The main goal of filtering is to obtain, recursively in time, good estimates of the state of a stochastic dynamical system based on noisy partial observations of the same. In settings where the signal/observation dynamics are significantly nonlinear or the noise intensities are high, an extended Kalman filter (EKF), which is essentially a first order approximation to an infinite dimensional problem, can perform quite poorly: it may require very frequent re-initializations and in some situations may even diverge. The theory of nonlinear filtering addresses these difficulties by considering the evolution of the conditional distribution of the state of the system given all the available observations, in the space of probability measures. We survey a variety of numerical schemes that have been developed in the literature for approximating the conditional distribution described by such stochastic evolution equations; with a special emphasis on an important family of schemes known as the particle filters. A numerical study is presented to illustrate that in settings where the signal/observation dynamics are non linear a suitably chosen nonlinear scheme can drastically outperform the extended Kalman filter.     

混沌信号在无线传感器网络中的盲分离

混沌信号在本质上属于非线性非高斯信号,它在无线传感器网络下的应用还涉及到信号量化问题,这使得混沌信号在此应用环境下的信号盲分离更为棘手.针对此问题,本文在容积卡尔曼粒子滤波的框架下提出一种解决方法.文中首先推导出观测信号的概率密度函数,在量化比特有限的情况下,采用最优量化器,获得最优的量化结果.在此基础上,使用容积卡尔曼滤波器产生粒子滤波中的重要性概率密度函数,融入最新的观测值,提高粒子对系统状态后验概率的逼近,提高信号盲分离的精度.仿真结果表明算法能够有效地分离混合混沌信号,参数估计的精度及其运算量均优于已有的无先导卡尔曼粒子滤波算法,其运行时间为无先导卡尔曼粒子滤波算法的88.77%.     

一种无线传感器网络中的混沌信号重构算法

将无线传感器网络节点观测区域中的一个混沌信号发送到融合中心,进行信号重构.由于节点的通信带宽受限,信号传输之前需要进行量化,给信号带来量化噪声,使得信号重构工作变得更为棘手.本文提出用平方根容积卡尔曼滤波器对融合中心收集的信号进行重构.首先估计观测信号的概率密度函数,使用最优量化器量化观测信号,在有限的量化比特数下,取得最优的信号量化性能.平方根容积卡尔曼滤波器相对无先导卡尔曼算法具有较少的求容积分点,因此具有计算量小的优点,同时迭代过程采用传递误差矩阵的平方根矩阵,保证迭代过程的稳定性和提高数据估计精度.仿真结果表明,该算法能够有效和快速地重构观测信号,并且比基于无先导卡尔曼滤波的算法更快.     

Variational Bayesian-Based Improved Maximum Mixture Correntropy Kalman Filter for Non-Gaussian Noise

The maximum correntropy Kalman filter (MCKF) is an effective algorithm that was proposed to solve the non-Gaussian filtering problem for linear systems. Compared with the original Kalman filter (KF), the MCKF is a sub-optimal filter with Gaussian correntropy objective function, which has been demonstrated to have excellent robustness to non-Gaussian noise. However, the performance of MCKF is affected by its kernel bandwidth parameter, and a constant kernel bandwidth may lead to severe accuracy degradation in non-stationary noises. In order to solve this problem, the mixture correntropy method is further explored in this work, and an improved maximum mixture correntropy KF (IMMCKF) is proposed. By derivation, the random variables that obey Beta-Bernoulli distribution are taken as intermediate parameters, and a new hierarchical Gaussian state-space model was established. Finally, the unknown mixing probability and state estimation vector at each moment are inferred via a variational Bayesian approach, which provides an effective solution to improve the applicability of MCKFs in non-stationary noises. Performance evaluations demonstrate that the proposed filter significantly improves the existing MCKFs in non-stationary noises.     

基于状态观测器的分数阶时滞混沌系统同步研究

研究分数阶时滞混沌系统同步问题,基于状态观测器方法和分数阶系统稳定性理论,设计分数阶时滞混沌系统同步控制器,使得分数阶时滞混沌系统达到同步,同时给出了数学证明过程.该同步控制器采用驱动系统和响应系统的输出变量进行设计,无需驱动系统和响应系统的状态变量,简化了控制器的设计,提高了控制器的实用性.利用Lyapunov稳定性理论和分数阶线性矩阵不等式,研究并给出了同步控制器参数的选择条件.以分数阶时滞Chen混沌系统为例,设计基于状态观测器的同步控制器,实现了分数阶时滞Chen混沌系统同步,并将其应用于保密通信系统中.仿真结果证明了该同步方法的有效性.     

A Nonlinear Maximum Correntropy Information Filter for High-Dimensional Neural Decoding

Neural signal decoding is a critical technology in brain machine interface (BMI) to interpret movement intention from multi-neural activity collected from paralyzed patients. As a commonly-used decoding algorithm, the Kalman filter is often applied to derive the movement states from high-dimensional neural firing observation. However, its performance is limited and less effective for noisy nonlinear neural systems with high-dimensional measurements. In this paper, we propose a nonlinear maximum correntropy information filter, aiming at better state estimation in the filtering process for a noisy high-dimensional measurement system. We reconstruct the measurement model between the high-dimensional measurements and low-dimensional states using the neural network, and derive the state estimation using the correntropy criterion to cope with the non-Gaussian noise and eliminate large initial uncertainty. Moreover, analyses of convergence and robustness are given. The effectiveness of the proposed algorithm is evaluated by applying it on multiple segments of neural spiking data from two rats to interpret the movement states when the subjects perform a two-lever discrimination task. Our results demonstrate better and more robust state estimation performance when compared with other filters.     

基于改进雁群PSO算法的模糊自适应扩展卡尔曼滤波的SLAM算法

针对扩展卡尔曼滤波同时定位与地图创建算法中难以建立准确的先验噪声模型的问题, 提出一种基于改进雁群粒子群算法的模糊自适应卡尔曼滤波算法. 利用分数阶微积分改进粒子进化速度, 利用混沌来改进粒子的初始化和发生早熟时的处理. 改进后的雁群粒子群算法在收敛速度与避免早熟方面有了很大改进, 并将改进的雁群粒子群算法用于模糊自适应扩展卡尔曼滤波同时定位与地图创建算法的训练, 并与用雁群粒子群算法训练的模糊自适应扩展卡尔曼滤波同时定位与地图创建算法进行对比, 其在定位与构图方面有很大的提高. 关键词： 同时定位与地图创建 雁群粒子群算法 分数阶微积分 混沌     

X荧光分析仪中数字基线估计的研究

在数字化X荧光分析仪中,不稳定的基线电压,会直接影响到仪器的性能,造成能量分辨率下降。基于此,利用卡尔曼滤波算法,对数字化后的X射线荧光信号进行基线估计。由于现有的经典卡尔曼滤波、简化sage-husa和改进sage-husa算法模型进行基线滤波的效果都不佳,因此有必要对现有的算法进行改进和优化。提出双重遗忘法,建立新型的基于sage-husa自适应卡尔曼滤波算法模型。实验结果表明,利用该数学模型进行基线滤波,取得了很好的滤波效果。避免了滤波发散和基线收敛缓慢的问题,实现了脉冲基线恢复,提高了仪器的能量分辨率。     

逼近积分点数下限的五阶容积卡尔曼滤波定轨算法

为了在保持滤波定轨精度不变的条件下提高定轨计算的实时性,提出一种新的逼近积分点个数下限的五阶容积卡尔曼滤波定轨算法.首先,采用一种数值容积准则对非线性函数的高斯加权积分进行近似,该准则所需的积分点个数仅比五阶代数精度容积准则积分点个数的理论下限多一个积分点,并在贝叶斯滤波算法框架下推导出本文算法的更新步骤.然后,给出实时定轨所需的状态方程和量测方程,在状态方程中考虑了J2项引力摄动和大气阻力摄动,在量测方程中利用坐标系转换推导了轨道状态与测量元素之间的非线性关系.仿真实验结果表明,本文所提算法在定轨精度方面与已有的五阶滤波算法相当,但所需的积分点个数最少,计算实时性最高,从而验证了本文算法的有效性.     

基于Kalman滤波算法的陀螺仪动态漂移补偿研究

应用MEMS陀螺仪测量人体手臂运动姿态时,针对陀螺仪受线加速度干扰导致测量姿态发散的问题,提出基于Kalman滤波算法的姿态误差补偿方法;该方法首先将陀螺仪采集到的角速度通过方向余弦算法解算得到姿态角,并将陀螺仪动态漂移造成的姿态角误差视为时变信号,通过建立姿态角漂移误差的状态方程及观测方程,应用卡尔曼滤波算法,实现对姿态角漂移误差的估计,最终达到对陀螺仪动态漂移误差的补偿;实验与仿真结果表明,应用该算法能够有效的抑制线加速度干扰导致的陀螺仪测量的姿态发散,适用于陀螺仪对人体手臂运动姿态的测量。     

Sequential Bayesian geoacoustic inversion for mobile and compact source-receiver configuration

Geoacoustic characterization of wide areas through inversion requires easily deployable configurations including free-drifting platforms, underwater gliders and autonomous vehicles, typically performing repeated transmissions during their course. In this paper, the inverse problem is formulated as sequential Bayesian filtering to take advantage of repeated transmission measurements. Nonlinear Kalman filters implement a random-walk model for geometry and environment and an acoustic propagation code in the measurement model. Data from MREA/BP07 sea trials are tested consisting of multitone and frequency-modulated signals (bands: 0.25-0.8 and 0.8-1.6 kHz) received on a shallow vertical array of four hydrophones 5-m spaced drifting over 0.7-1.6 km range. Space- and time-coherent processing are applied to the respective signal types. Kalman filter outputs are compared to a sequence of global optimizations performed independently on each received signal. For both signal types, the sequential approach is more accurate but also more efficient. Due to frequency diversity, the processing of modulated signals produces a more stable tracking. Although an extended Kalman filter provides comparable estimates of the tracked parameters, the ensemble Kalman filter is necessary to properly assess uncertainty. In spite of mild range dependence and simplified bottom model, all tracked geoacoustic parameters are consistent with high-resolution seismic profiling, core logging P-wave velocity, and previous inversion results with fixed geometries.     

光子计数激光雷达的时间相关卡尔曼深度估计

在高背景噪声和低积分时间的激光雷达远距离成像场景中,针对传统方法得到的深度图像目标被噪声淹没和深度估计偏差较大的问题,提出了一种基于信号光子时间相关性和自适应卡尔曼滤波器的深度信息估计方法。首先,提取在时间上具有聚集特征的光子计数形成集合;然后,分析了影响信号光子在时间上分布的因素并使用静态高斯线性模型来描述该集合;最后将集合中的所有光子飞行时间乱序,输入改进的自适应卡尔曼滤波器,从而迭代估计深度值。在信号噪声比为1的室内,积分时间分别为10 ms和1 ms时,本文方法相对传统的最大似然方法在均方根误差指标上提升了40%和38%。在信噪比约为0.135的室外2 km目标成像实验中,在信号光子数分别为100、33和17的情况下,本文方法成像效果都优于传统最大似然估计方法和时间相关光子快速去噪方法,得到的深度图像都更清晰,噪声更低。在高噪声和短积分时间下,本文方法可以被运用于激光雷达远距离成像的深度信息估计和图像恢复中。     

Polarization de-multiplexing using a modified Kalman filter in CO-OFDM transmissions

We propose the modified Kalman filter(MKF) using the received signal for observation and constructing an inverse process of the conventional Kalman filter(CKF) for polarization de-multiplexing in coherent optical(CO) orthogonal frequency-division multiplexing(OFDM) transmissions. The MKF can avoid the convergence error problem in CKF without matrix inverse operation and has a faster converging speed and a much larger tolerance to the process and measurement noise covariance, about two orders of magnitude more than those of CKF. We experimentally demonstrate the 12 Gbaud OFDM signal transmission over 480 km standard singlemode fiber. The performance of MKF and CKF outperforms pilot-aided polarization de-multiplexing with better accuracy and nonlinearity tolerance.     

A Maximum Entropy Method for Particle Filtering

Standard ensemble or particle filtering schemes do not properly represent states of low priori probability when the number of available samples is too small, as is often the case in practical applications. We introduce here a set of parametric resampling methods to solve this problem. Motivated by a general H-theorem for relative entropy, we construct parametric models for the filter distributions as maximum-entropy/minimum-information models consistent with moments of the particle ensemble. When the prior distributions are modeled as mixtures of Gaussians, our method naturally generalizes the ensemble Kalman filter to systems with highly non-Gaussian statistics. We apply the new particle filters presented here to two simple test cases: a one-dimensional diffusion process in a double-well potential and the three-dimensional chaotic dynamical system of Lorenz.     

Performance analysis of dispersion compensating optical FIR filters designed using a quasi-analytic synthesis approach

In this paper we investigate the performance of a quasi-analytic synthesis approach for dispersion compensating optical finite impulse response filters. It is shown by eye opening penalty and optical signal to noise ratio penalty calculations that with this method the filters dispersion can be adjusted reliably and with low effort to compensate for a specific amount of residual chromatic dispersion in optical transmission systems. The maximum attainable dispersion is then only limited by the filter order and the free spectral range rather then by optimization problems within the filter design.     

基于一组Kalman滤波器信息融合的故障诊断

为了能及时准确地诊断发动机的传感器和执行机构故障,本文提出了基于一组Kalman滤波器信息融合的方法进行故障诊断。首先根据传感器特性设计了一组滤波器用于传感器故障诊断、隔离,每个滤波器针对一个传感器进行设计;其次根据执行机构故障特性设计了一组Kalman滤波器进行执行机构偏差估计,从而对执行机构进行故障诊断、隔离;接着给出了传感器、执行机构信息融合的诊断方案;最后分别给传感器、执行机构添加故障进行方案验证,仿真结果得出在传感器或者执行机构任意部件出故障的情况下,该融合方法可以有效地诊断并隔离出有故障的传感器或者执行机构。     

基于Huber的高阶容积卡尔曼跟踪算法

为改善高阶容积卡尔曼滤波算法的滤波精度和鲁棒性, 提出了一种新的基于Huber的高阶容积卡尔曼滤波算法. 在采用统计线性回归模型近似非线性量测模型的基础上, 利用Huber M 估计算法实现状态的量测更新. 进一步结合高阶球面-径向容积准则的状态预测模块构成基于 Huber的高阶容积卡尔曼跟踪算法. 重点分析了Huber代价函数的调节因子对算法跟踪性能的影响. 通过对纯方位目标跟踪和再入飞行器跟踪两个实例验证了所提算法的跟踪性能优于传统高阶容积卡尔曼滤波算法.     

A physically realizable whitening filter for cryogenic resonant gravitational-wave antennas

In this paper the problem of a physically realizable whitening-filter (WF) synthesis for cryogenic resonant gravitational-wave antennas is considered in the framework of the optimal linear Kalman-Bucy filtering. A system of equations determining the structure of the Kalman filter for gravitational-wave antennas with a displacement converter is presented. The transfer function of a stationary physically realizable WF is derived. The Kalman approach ensures the possibility of high frequency parameter measurements of Gaussian and non-Gausssian noises in low-dissipation oscillatory systems for observation intervals, which are relatively short in comparison to the relaxation time.     

光电跟踪系统的共轴跟踪控制技术研究

为了提高光电跟踪系统的跟踪精度,对其粗跟踪环节的共轴跟踪控制技术进行了研究。介绍了共轴跟踪控制的基本原理,分析了常用非线性卡尔曼滤波算法,并仿真比较了无迹卡尔曼滤波(UKF)和容积卡尔曼滤波(CKF)的位置和速度的预测精度。在此基础上综合UKF和CKF的优点,设计了对延迟的合成位置信号进行处理的双并联滤波器,实现了光电跟踪系统的共轴跟踪控制。利用实测数据仿真实验表明,光电跟踪系统的跟踪精度明显提高。     

Filtering skill for turbulent signals for a suite of nonlinear and linear extended Kalman filters

The filtering skill for turbulent signals from nature is often limited by errors due to utilizing an imperfect forecast model. In particular, real-time filtering and prediction when very limited or no a posteriori analysis is possible (e.g. spread of pollutants, storm surges, tsunami detection, etc.) introduces a number of additional challenges to the problem. Here, a suite of filters implementing stochastic parameter estimation for mitigating model error through additive and multiplicative bias correction is examined on a nonlinear, exactly solvable, stochastic test model mimicking turbulent signals in regimes ranging from configurations with strongly intermittent, transient instabilities associated with positive finite-time Lyapunov exponents to laminar behavior. Stochastic Parameterization Extended Kalman Filter (SPEKF), used as a benchmark here, involves exact formulas for propagating the mean and covariance of the augmented forecast model including the unresolved parameters. The remaining filters use the same nonlinear forecast model but they introduce model error through different moment closure approximations and/or linear tangent approximation used for computing the second-order statistics of the augmented stochastic forecast model. A comprehensive study of filter performance is carried out in the presence of various moment closure errors which are enhanced by additional model errors due to incorrect parameters inducing additive and multiplicative stochastic biases. The estimation skill of the unresolved stochastic parameters is also discussed and it is shown that the linear tangent filter, despite its popularity, is completely unreliable in many turbulent regimes for both parameter estimation and filtering; moreover, regimes of filter divergence for the linear tangent filter are identified. The results presented here provide useful guidelines for filtering turbulent, high-dimensional, spatially extended systems with more general model errors, as well as for designing more skillful methods for superparameterization of unresolved intermittent processes in complex multi-scale models. They also provide unambiguous benchmarks for the capabilities of linear and nonlinear extended Kalman filters using incorrect statistics on an exactly solvable test bed with rich and realistic dynamics.