Algorithm of gaussian sum filter based on high-order UKF for dynamic state estimation

In this work we consider the state estimation problem in nonlinear/non-Gaussian systems. A new version of Gaussian sum estimation algorithm is developed here based on high-order unscented Kalman filter (HUKF). A sigma point selection method, high-order unscented transformation (HUT) technique is proposed for the HUKF, which can approximate the Gaussian distributions more accurately. We present the systematic formulation of Gaussian filters and develop efficient and accurate numerical integration of the optimal filter. We then go on to extend the use of the HUKF to discrete-time, nonlinear systems with additive, possibly non-Gaussian noise. The resulting filtering algorithm, called the Gaussian sum high-order unscented Kalman filter (GS-HUKF) approximates the predicted and posterior densities as a finite number of weighted sums of Gaussian densities. It is corroborated in the theoretical analysis and the simulation that the proposed Gaussian sum HUKF has integrated advantages with respect to computational accuracy and time complexity for nonlinear non-Gaussian filtering problems.     

State estimation of nonlinear dynamical systems using nonlinear update based Unscented Gaussian Sum Filter

Two attractive features of Unscented Kalman Filter (UKF) are: (1) use of deterministically chosen points (called sigma points), and (2) only a linear dependence of the number of sigma points on the number of states. However, an implicit assumption in UKF is that the prior conditional state probability density and the state and measurement noise densities are Gaussian. To avoid the restrictive Gaussianity assumption, Gaussian Sum-UKF (GS-UKF) has been proposed in literature that approximates all the underlying densities using a sum of Gaussians. However, the number of sigma points required in this approach is significantly higher than in UKF, thereby making GS-UKF computationally intensive. In this work, we propose an alternate approach, labeled Unscented Gaussian Sum Filter (UGSF), for state estimation of nonlinear dynamical systems, corrupted by Gaussian state and measurement noises. Our approach uses a Sum of Gaussians to approximate the non-Gaussian prior density. A key feature of this approximation is that it is based on the same number of sigma points as used in UKF, thereby resulting in similar computational complexity as UKF. We implement the proposed approach on two nonlinear state estimation case studies and demonstrate its utility by comparing its performance with UKF and GS-UKF.     

基于均差滤波与高斯和的非线性非高斯系统滤波算法

针对一类非线性非高斯系统的滤波问题,在分析均差滤波算法和高斯和滤波算法的基础上,提出一种基于均差滤波的高斯和滤波算法,适于处理非线性非高斯系统的滤波问题.对于似然密度位于条件转移概率密度拖尾处的情况,与传统的粒子滤波算法相比,所提算法能提高滤波的精度和实时性.仿真实验验证了新算法的有效性.     

A new real-time suboptimum filtering and prediction scheme for general nonlinear discrete dynamic systems with Gaussian or non-Gaussian noise

A new suboptimum state filtering and prediction scheme is proposed for nonlinear discrete dynamic systems with Gaussian or non-Gaussian disturbance and observation noises. This scheme is an online estimation scheme for real-time applications. Furthermore, this scheme is very suitable for state estimation under either constraints imposed on estimates or missing observations. State and observation models can be any nonlinear functions of the states, disturbance and observation noises as long as noise samples are independent, and the density functions of noise samples and conditional density functions of the observations given the states are available. State models are used to calculate transition probabilities from gates to gates. If these transition probabilities are known or can be estimated, state models are not needed for estimation. The proposed scheme (PR) is based upon state quantisation and multiple hypothesis testing. Monte Carlo simulations have shown that the performance of the PR, sampling importance resampling (SIR) particle filter and extended Kalman (EK) filter are all model-dependent, and that the performance of the PR is better than both the SIR particle filter and EK filter for some nonlinear models, simulation results of three of which are given in this article.     

适用处理非高斯观测噪声的强跟踪卡尔曼滤波器

在高斯噪声条件下,卡尔曼滤波器(KF)能够获得系统状态的一致最小方差线性无偏估计.但当噪声非高斯,KF性能将严重下降.观测噪声非高斯现象在深空探测自主导航中经常遇到,然而现有模型可能存在着精度不高、稳定性不强或者计算复杂度较高的缺点.针对这种现状,本文在传统强跟踪卡尔曼滤波器(STKF)中新息正交原则的基础上,推导了适用处理非高斯观测噪声的强跟踪卡尔曼滤波器(STKFNO),并将其嵌入到无迹卡尔曼滤波(UKF)框架下形成适用处理非线性系统非高斯观测噪声的强跟踪无迹卡尔曼滤波器(STUKFNO).所提出的算法被应用到深空光学自主导航系统中,仿真结果表明所提出的算法能够较好地应对观测噪声的非高斯性.     

Algorithm of Gaussian Sum Filter Based on SGQF for Nonlinear Non-Gaussian Models

To improve the filtering effect of the sparse grid quadrature filter (SGQF) under non-Gaussian conditions, the Gaussian sum technique is introduced, and the Gaussian sum sparse grid quadrature filter (GSSGQF) is developed. We present a systematic formulation of the SGQF and extend it to the discrete-time nonlinear system with the non-Gaussian noise. The proposed algorithm approximates the non-Gaussian probability densities by a finite number of weighted sums of Gaussian densities, and takes the SGQF as the Gaussian sub-filter to conduct the time and measurement update for each Gaussian component. An application in the discrete-time nonlinear system with the non-Gaussian noise has been shown to demonstrate the accuracy of the GSSGQF. It outperforms the unscented Kalman filter (UKF), the cubature Kalman filter (CKF) and the SGQF. Theoretical analysis and simulation results prove that the GSSGQF provides significant performance improvement in the calculation accuracy for nonlinear non-Gaussian filtering problems.

     

New theoretical framework for OFDM/CDMA systems with peak-limited nonlinearities

A new theoretical framework for the evaluation of the in-band nonlinear distortion effects on the performance of OFDM systems is presented. In contrast to previous works that approximate the nonlinear noise as a Gaussian additive random process, the new framework is based on the properties of the large deviations of a stationary Gaussian process and shot noise theories, which can evaluate the performance of the OFDM system with high accuracy, especially at realistic scenarios where the Gaussian approximation of the nonlinear noise is no longer valid. The approach can be used to evaluate many communication systems with peak-limited nonlin-earities and high PAPR, such as the downlink performance analysis of large ca-pacity DS-CDMA systems.     

UKF、PF与UPF跟踪性能的比较

无迹卡尔曼滤波器(UKF)是用一系列确定样本来逼近状态的后验概率密度,对任何非线性高斯系统都有较好的跟踪性能。粒子滤波器(PF)是用随机样本来近似状态后验概率密度函数,适用于任何非线性非高斯系统,但当似然函数出现在转移概率密度函数的尾部或者在高精度测量的场合,PF的跟踪性能降低。针对强非线性、非高斯系统、高精度测量的环境,文中提出采用UPF算法进行跟踪,并对PF、UKF和UPF三种跟踪算法进行了仿真,结果表明,UPF的跟踪精度要远高于PF、UKF的精度。     

A dynamic programming approach to trajectory estimation

An iterative equation based on dynamic programming for finding the most likely trajectory of a dynamic system observed through a noisy measurement system is presented; the procedure can be applied to nonlinear systems with non-Gaussian noise. It differs from the recently developed Bayesian estimation procedure in that the most likely estimate of the entire trajectory up to the present time, rather than of the present state only, is generated. It is shown that the two procedures in general yield different estimates of the present state; however, in the case of linear systems with Gaussian noise, both procedures reduce to the Kalman-Bucy filter. Illustrative examples are presented, and the present procedure is compared with the Bayesian procedure and with other estimation techniques in terms of computational requirements and applicability.     

基于SRCKF 的自适应高斯和状态滤波算法

针对非线性非高斯离散动态系统中的状态估计问题,基于高斯和递推关系,提出一种高斯和状态估计算法GSSRCKF.首先将状态噪声、观测噪声及滤波初值均表示为高斯和的形式,以平方根容积卡尔曼滤波为子滤波器分别估计各高斯子项对应的系统状态;然后结合各子项对应的权值实现全局估计;最后设计高斯子项对应权值的自适应策略,并采用约简控制法降低计算复杂度.仿真结果验证了所提出的算法在滤波稳定性方面的优越性.     

非高斯噪声下的回声消除算法研究

回声消除一直是信号处理领域的热门研究方向,其中自适应滤波器是在回声消除问题中最为广泛应用的技术,但自适应滤波算法主要是在基于高斯噪声条件下的应用,而现实环境广泛存在着非高斯的噪声,这严重影响了基于L2范数的自适应噪声滤波算法的噪声消除性能。为解决回声消除方法对非高斯噪声的适用性问题,根据回声路径具有明显的稀疏系统特性,结合比例矩阵的设计思想以及符号算法(SA),提出一种改进的MIPNSA算法。该滤波算法既能很好地适应于不同的背景噪声,同时也在较大程度上增强了对稀疏系统的适应能力。仿真测试结果表明,在高斯噪声和非高斯噪声条件下,本算法比现有的一些算法的回声消除效果更佳。     

基于高斯和近似的扩展切片高斯混合滤波器及其在多径估计中的应用

全球卫星导航系统(Global navigation satellite system, GNSS)信号的多径估计问题实际上是条件线性状态空间模型下的状态估计问题. 根据高斯和理论提出了适用于非高斯噪声环境的扩展切片高斯混合滤波(Extension of sliced Gaussian mixture filter, ESGMF)算法. 该算法将非高斯噪声的状态概率密度函数(Probability density function, PDF)表示为高斯和的形式,将ESGMF通过一组并行的切片高斯混合滤波器(Sliced Gaussian mixture filter, SGMF)来实现.同时, 在ESGMF算法中利用粒子滤波(Particle filter, PF)中重采样的思想对成指数增加的状态预测PDF的高斯混合个体进行约简, 以提高贝叶斯推理的效率.该算法可以获得非高斯噪声下状态PDF的迭代解析表达式. 最后, 将ESGMF应用于GPS多径参数估计, 仿真结果表明, ESGMF算法的估计精度优于基于PF和扩展卡尔曼滤波(Extended Kalman filter, EKF)的算法.     

An adaptive robustizing approach to kalman filtering

The performance of a linear Kalman filter will degrade when the dynamic noise is not Gaussian. A robust Kalman filter based on the m-interval polynomial approximation (MIPA) method for unknown non-Gaussian noise is proposed. Two situations are considered: (a) the state is Gaussian and the observation noise is non-Gaussian; (b) the state is non-Gaussian and the observation noise is Gaussian. It is shown, as compared with other non-Gaussian filters, the MIPA Kalman filter is computationally feasible, unbiased, more efficient and robust. For the scalar model, Monte Carlo simulations are given to demonstrate the ideas involved.     

Particle filtering-based fault detection in non-linear stochastic systems

Much of the development in model-based fault detection techniques for dynamic stochastic systems has relied on the system model being linear and the noise and disturbances being Gaussian. Linearized approximations have been used in the non-linear systems case. However, linearization techniques, being approximate, tend to suffer from poor detection or high false alarm rates. A novel particle filtering based approach to fault detection in non-linear stochastic systems is developed here. One of the appealing advantages of the new approach is that the complete probability distribution information of the state estimates from particle filter is utilized for fault detection, whereas, only the mean and covariance of an approximate Gaussian distribution are used in a coventional extended Kalman filter-based approach. Another advantage of the new approach is its applicability to general non-linear system with non-Gaussian noise and disturbances. The effectiveness of this new method is demonstrated through Monte Carlo simulations and the detection performance is compared with that using the extended Kalman filter on a non-linear system.     

一种新的高动态GPS载波跟踪算法

提出了基于无迹粒子滤波（UPF）算法的高动态GPS载波跟踪环路,仿真分析了该方案在高斯噪声和非高斯噪声环境下对高动态GPS信号的跟踪性能,并与分别基于扩展卡尔曼滤波（EKF）、无迹卡尔曼滤波（UKF）、粒子滤波（PF）及扩展卡尔曼粒子滤波（EPF）这四种算法的载波跟踪环路进行了性能对比。仿真结果表明,基于UPF估计器的载波跟踪环路在高动态、弱信号以及非高斯噪声环境下具有优越的跟踪性能,既可以提高跟踪精度,又解决了非高斯噪声干扰问题。通过模拟实验验证了该方案的有效性。     

自适应高斯粒子滤波在导航中的应用

针对粒子滤波中重要性密度函数难以选取和粒子退化导致的计算精度下降的问题,提出一种新的自适应高斯粒子滤波算法。通过高斯混合密度函数和UT变换来获取状态均值和协方差阵,选择并计算合适的自适应因子来调节均值和方差,在迭代过程中可动态调节重要性密度函数,并用WEM和EM步骤代替重采样,上述滤波算法考虑了最新量测信息的影响,使滤波性能明显改善,能更好地解决非线性非高斯系统模型的抗干扰问题。将提出的算法应用于SINS/GPS组合导航系统跑车试验中,结果表明上述滤波算法能提高导航解算的精度,其性能明显优于已有滤波,同时验证了当系统出现噪声干扰突然变化时提出算法的有效性。     

Identification and prediction of a class of non-Gaussian time-series via transformation

Whereas optimal prediction of Gaussian sequences requires the employment of a linear filter with consistently identifiable parameters and with Gaussian white noise input, the optimal predictor of non-Gaussian sequences is n nonlinear filter, having an independent noise input. Since the latter cannot be identified directly without prior knowledge of the non-linearity, the optimal linear predictor is usually identified where a non-Gaussian white noise input is considered and which is fully optimal only when that input turns out to be independent in all moments. However, if the non-Gaussian sequence is the outcome of a Gaussian sequence passed through a zero memory non-linearity or through non-linear measurement elements, a transformation of the non-Gaussian sequence into a Gaussian one is possible, such that optimal non-linear prediction may be approximated to any required degree, as is shown by the analysis of the present work. Furthermore, the parameters of that predictor may be consistently identified in the absence of any parameter information.     

鲁棒高斯和集合卡尔曼滤波及其在纯角度跟踪中的应用

针对纯角度目标跟踪中量测信息易受异常值和非高斯噪声干扰的问题,提出了一种新的非线性滤波算法–鲁棒高斯和集合卡尔曼滤波(robust Gaussian-sum ensemble Kalman filter,RGSEnKF)算法.首先,采用Huber技术重塑集合卡尔曼滤波的量测更新过程,能够有效地处理量测中的异常值.随后,将改进的集合卡尔曼滤波在高斯和框架下进行扩展,得到RGSEnKF算法,可以进一步解决受非高斯噪声干扰的非线性系统的状态估计问题.此外,新算法中包含距离参数化初始化策略和高斯分量融合策略.前者是为了减小纯角度跟踪中距离信息不可观测的影响,而后者可以避免高斯分量数目随时间不断增长.大量仿真结果验证了新算法的有效性和鲁棒性.     

一种新的非线性目标跟踪方法

在实际的目标跟踪场景中,普遍存在非高斯过程噪声和/或量测噪声,以及非高斯先验信息等情况,针对这一问题,提出一种新的解决非线性/非高斯系统滤波问题的非线性滤波算法,即高斯和求积分卡尔曼滤波（GSQKF）算法。仿真实验将新算法与标准的粒子滤波算法进行了比较,表明新算法是一种非常有效的非线性滤波算法。     

Nonlinear Filtering of Non-Gaussian Noise

This paper introduces a new nonlinear filter for a discrete time, linear system which is observed in additive non-Gaussian measurement noise. The new filter is recursive, computationally efficient and has significantly improved performance over other linear and nonlinear schemes. The problem of narrowband interference suppression in additive noise is considered as an important example of non-Gaussian noise filtering. It is shown that the new filter outperforms currently used approaches and at the same time offers simplicity in the design.