Noise covariance matrix estimation with subspace model identification for Kalman filtering

A problem frequently encountered in Kalman filtering is the tuning of the noise covariance matrices. Indeed, misspecifying their values can drastically reduce the performance of the Kalman filter. Unfortunately, in most practical cases, noise statistics are not known a priori. This paper focuses on a method relying on subspace model identification theory to determine them accurately. This solution is developed for linear time invariant systems with stationary random disturbances having constant covariance matrices. Practically, these noise covariance matrices are determined from the comparison between an estimated state space representation and the discrete time state space representation involved in the Kalman filter equations. The method developed in this paper departs from most of the solutions available in the literature by the fact that it does not need any tuning parameter to be chosen by the user. After discussing theoretical results, several numerical examples are given to demonstrate the efficiency of the approach.     

Optimal and self‐tuning fusion Kalman filters for discrete‐time stochastic singular systems

Based on the optimal fusion estimation algorithm weighted by scalars in the linear minimum variance sense, a distributed optimal fusion Kalman filter weighted by scalars is presented for discrete‐time stochastic singular systems with multiple sensors and correlated noises. A cross‐covariance matrix of filtering errors between any two sensors is derived. When the noise statistical information is unknown, a distributed identification approach is presented based on correlation functions and the weighted average method. Further, a distributed self‐tuning fusion filter is given, which includes two stage fusions where the first‐stage fusion is used to identify the noise covariance and the second‐stage fusion is used to obtain the fusion state filter. A simulation verifies the effectiveness of the proposed algorithm. Copyright © 2008 John Wiley & Sons, Ltd.     

基于自适应容积卡尔曼滤波算法的电力系统动态谐波状态估计

由于传统的谐波状态估计的参数辨识算法要求噪声的协方差矩阵固定不变,而实际工程中噪声的协方差矩阵是随时间变化的,工程中存在错误的量测数据,导致传统参数辨识算法估计的谐波电流参数的准确度较低。因此,提出自适应容积卡尔曼滤波算法来提高辨识谐波电流参数的准确度。首先,针对时变噪声干扰,采用基于渐消记忆指数加权法的噪声估值器算法生成时变噪声的协方差矩阵;其次,针对错误的量测数据,采用开窗估计算法修正错误的量测数据;然后,将修正的噪声协方差矩阵和量测数据代入容积卡尔曼滤波算法中,对谐波电流参数进行估计;最后,搭建IEEE 13节点系统仿真模型,验证了自适应容积卡尔曼滤波算法在时变噪声干扰及量测数据错误情况下仍可准确地估计谐波电流参数,确保了动态谐波状态估计的准确性。     

Process noise covariance estimation via stochastic approximation

Kalman filtering for linear systems is known to provide the minimum variance estimation error, under the assumption that the model dynamics is known. While many system identification tools are available for computing the system matrices from experimental data, estimating the statistics of the output and process noises is still an open problem. Correlation-based approaches are very fast and sufficiently accurate, but there are typically restrictions on the number of noise covariance elements that can be estimated. On the other hand, maximum likelihood methods estimate all elements with high accuracy, but they are computationally expensive, and they require the use of external optimization solvers. In this paper, we propose an alternative solution, tailored for process noise covariance estimation and based on stochastic approximation and gradient-free optimization, that provides a good trade-off in terms of performance and computational load, and is also easy to implement. The effectiveness of the method as compared to the state of the art is shown on a number of recently proposed benchmark examples.     

Robust adaptive unscented Kalman filter for attitude estimation of pico satellites

Unscented Kalman filter (UKF) is a filtering algorithm that gives sufficiently good estimation results for the estimation problems of nonlinear systems even when high nonlinearity is in question. However, in case of system uncertainty or measurement malfunctions, the UKF becomes inaccurate and diverges by time. This study introduces a fault‐tolerant attitude estimation algorithm for pico satellites. The algorithm uses a robust adaptive UKF, which performs correction for the process noise covariance (Q‐adaptation) or measurement noise covariance (R‐adaptation) depending on the type of the fault. By the use of a newly proposed adaptation scheme for the conventional UKF algorithm, the fault is detected and isolated, and the essential adaptation procedure is followed in accordance with the fault type. The proposed algorithm is tested as a part of the attitude estimation algorithm of a pico satellite. Copyright © 2013 John Wiley & Sons, Ltd.     

基于扩展卡尔曼滤波器的感应电机转子断条故障检测

根据扩展卡尔曼滤波器的原理,推导出用感应电机端电压、线电流信号估计电机转速的方法,还对扩展卡尔曼滤波器的参数进行了优化,提高了辨识的精度和收敛的速度.对电机转速进行FFT频谱分析,并以此为基础进行电机转子断条故障的在线检测.该方法不仅较以往电流检测方法更有利于克服边频带效应带来的影响,同时也避免了精密的转速测量装置的使用,因此具有广泛的实用价值.实测结果表明该方法是正确有效的.     

基于自适应CKF的锂离子电池SOC估算

扩展卡尔曼滤波(EKF)和无迹卡尔曼滤波(UKF)算法估算电池荷电状态(SOC)依赖等效模型参数的准确性,估算精度低。容积卡尔曼滤波(CKF)算法的滤波性能良好。利用自适应CKF(ACKF)算法估算电池SOC,自适应调节过程噪声协方差和量测噪声协方差,提高估算SOC的精度。对锂离子电池建立二阶RC等效电路模型,在不同工况下进行充放电,用卡尔曼滤波算法在线辨识等效模型的参数,ACKF算法实时估算SOC。ACKF算法估算SOC的鲁棒性较强,精度在1.5%以内。     

基于平方根容积卡尔曼滤波的发电机动态状态估计

发电机动态状态估计是电力系统动态安全分析的重要内容。针对容积卡尔曼滤波(CKF)在迭代中协方差阵不对称或非正定导致的估计精度下降甚至滤波发散问题,利用平方根滤波(SRF)能确保协方差阵非负定和数值稳定性方面的优势,提出基于平方根容积卡尔曼滤波(SRCKF)的发电机动态状态估计方法,并给出了计算步骤。最后,利用仿真系统和实际系统比较了SRCKF、CKF和无迹卡尔曼滤波(UKF)三种算法的估计性能,证明了SRCKF算法能够解决CKF滤波中因协方差阵非正定导致的滤波发散问题;同时SRCKF算法在计算效率、滤波精度和数值稳定性方面均优于CKF和UKF算法。     

Accurate tracking of harmonic signals in VSC-HVDC systems using PSO based unscented transformation

This paper presents the estimation of harmonics in a voltage source converter based HVDC (VSC-HVDC) system for designing AC side filters. The extended Kalman filter (EKF) is well known for estimating amplitude, phase, frequency, and harmonic content of a signal corrupted with noise. However, the EKF algorithm suffers from instability due to linearization and costly calculation of Jacobian matrices, and its performance deteriorates when the signal model is highly nonlinear. This paper, therefore, proposes an unscented Kalman filter (UKF) to overcome these difficulties of linearization and derivative calculations for robust tracking of harmonics in VSC-HVDC system. The model and measurement error covariance matrices Q and R along with the UKF parameters are selected using a modified particle swarm optimization (PSO) algorithm. To circumvent the problem of premature convergence and local minima, a dynamically varying inertia weight based on the variance of the population fitness is used. This results in a better local and global searching ability of the particles, which improves the convergence of the velocity and better accuracy of the UKF parameters. Various simulation results for harmonic signals corrupted with noise obtained from VSC-HVDC system reveal significant improvement in noise rejection and speed of convergence and accuracy.     

Exponential convergence of the Kalman filter based parameter estimation algorithm

In this paper we shall present a new method to analyse the convergence property of the Kalman filter based parameter estimation algorithms. This method for convergence analysis is mainly based on some matrix inequalities and is more simple than some of the existing approaches in the literature. This method can simultaneously provide both lower and upper bounds on the exponential convergence rate as the functions of bounds of the related matrices, such as the covariance matrices. A simulation example is provided to illustrate the convergence property of the Kalman filter based algorithms. Copyright © 2003 John Wiley & Sons, Ltd.     

永磁同步电机无传感器调速系统中EKF参数的调试

采用扩展卡尔曼滤波器(EKF)在估算永磁同步电动机(PMSM)转子位置以及转速时,协方差参数的确定一直是一个难点,必须通过多次仿真实验才可以确定,本文提出了一种基于标幺制的参数调试方法,只需要在一定范围内调试两个参数,即可确定协方差矩,阵保证系统稳定工作.     

Estimation of noise covariance matrices for periodic systems

Estimation of the noise covariance matrices for linear time‐variant stochastic dynamic periodic systems is treated. The novel offline method for estimation of the covariance matrices of the state and measurement noises is designed. The method is based on analysis of second‐order statistics of the state estimate produced by the linear multi‐step predictor. The estimates of the noise covariance matrices are unbiased and converge to the true values with increasing number of data. The theoretical results are illustrated in numerical examples. Copyright © 2011 John Wiley & Sons, Ltd.     

Noise covariance matrices in state‐space models: A survey and comparison of estimation methods—Part I

This paper deals with the estimation of the noise covariance matrices of systems described by state‐space models. Stress is laid on the systematic survey and classification of both the recursive and batch processing methods proposed in the literature with a special focus on the correlation methods. Besides the correlation methods, representatives of other groups are introduced also with respect to their basic idea, estimate properties, assumptions and possible extensions, and user‐defined parameters. Common and dual properties of the methods are highlighted, and a simulation comparison using exemplary MATLAB implementations of the methods is provided.     

An improved adaptive unscented Kalman filter for estimating the states of in‐wheel‐motored electric vehicle

Vehicle state is essential for active safety stability control. However, the accurate measurement of some vehicle states is difficult to achieve without the use of expensive equipment. To improve estimation accuracy in real time, this paper proposes an estimator of vehicle velocity based on the adaptive unscented Kalman filter (AUKF) for an in‐wheel‐motored electric vehicle (IWMEV). Given the merits of an independent drive structure, the tire forces of the IWMEV can be directly calculated through a vehicle dynamic model. Additionally, by means of the normalized innovation square, the validity of vehicle velocity estimation can be detected, and the sliding window length can be adjusted adaptively; thus, the steady‐state error and the dynamic performance of the IWMEV are demonstrated to be simultaneously improved over an alternative approach in comparisons. Then, an adaptive adjustment strategy for the noise covariance matrices is introduced to overcome the impact of parameter uncertainties. The numerically simulated and experimental results prove that the proposed vehicle velocity estimator based on AUKF not only improves estimation accuracy but also possesses strong robustness against parameter uncertainties. The deployment of the estimation algorithm by using a single‐chip microcomputer verifies the strong real‐time performance and easy‐to‐implement characteristics of the proposed algorithm.     

Effects of the weighting matrix on power system state estimation

The effects of the weighting matrices used in the weighted least squares (WLS) method when applied to the problem of power system state estimation (PSSE) are studied. In particular, the effects of two weighting matrices are examined: the first is the inverse of the covariance matrix of the measurement errors and the second is the one used in the utility PSSE programs. Other weighting matrices are also tested.Different performance indices for state comparison are defined. The results obtained by performing many Monte Carlo trials using these weighting matrices are compared. Results show that the inverse of the covariance matrix gives the best results.     

考虑参数不确定性影响的发电机动态状态估计方法

噪声统计特性和模型参数的不确定性,会严重影响动态状态估计的精度。针对该问题,文中提出了一种基于H∞容积卡尔曼滤波(HCKF)的动态状态估计新方法。首先,建立发电机动态状态估计模型;其次,依据H∞滤波理论构造模型不确定性约束准则,并在容积卡尔曼滤波(CKF)中依据该准则计算更新估计误差协方差阵,抑制参数不确定性对状态估计精度的影响;最后,通过对IEEE 10机39节点系统和某实际大区域电网系统的算例测试,将所提方法与CKF方法和改进插值扩展卡尔曼滤波(IEKF)方法的估计性能进行对比。算例仿真结果表明,HCKF方法在估计精度和对模型不确定性的鲁棒性方面较CKF和IEKF方法均有所提高,能够有效抑制模型不确定性对发电机动态状态估计的影响。     

考虑量测相关性的容积卡尔曼滤波动态状态估计

针对电力系统动态状态估计中SCADA量测量间存在相关性的实际情况,文中提出了一种考虑量测相关性的容积卡尔曼滤波动态状态估计方法。首先进行了SCADA量测相关性分析,然后基于状态转移方程推导过程噪声协方差矩阵,基于容积变换方法计算考虑SCADA量测相关性的量测误差协方差矩阵,并提出了考虑量测相关性的电力系统动态状态估计流程,每次估计实时修正量测误差协方差矩阵及过程噪声协方差矩阵。IEEE-39节点系统的仿真结果表明,相较于不考虑量测相关性的容积卡尔曼滤波算法,文中方法能够明显提高状态估计结果的精度。     

Distributed fusion white noise deconvolution estimators

The white noise deconvolution or input white noise estimation problem has important applications in oil seismic exploration, communication and signal processing. By combining the Kalman filtering method with the modern time series analysis method, based on the autoregressive moving average (ARMA) innovation model, new distributed fusion white noise deconvolution estimators are presented by weighting local input white noise estimators for general multisensor systems with different local dynamic models and correlated noises. The new estimators can handle input white noise fused filtering, prediction and smoothing problems, and are applicable to systems with colored measurement noise. Their accuracy is higher than that of local white noise deconvolution estimators. To compute the optimal weights, the new formula for local estimation error cross-covariances is given. A Monte Carlo simulation for the system with Bernoulli-Gaussian input white noise shows their effectiveness and performance.     

基于改进Att-LSTNet与无迹粒子滤波融合的主动配电网预测辅助状态估计

针对传统的无迹粒子滤波(unscented particle filter, UPF)存在不准确的新息向量及未知的量测噪声协方差矩阵导致估计精度低的问题,提出一种改进Att-LSTNet与UPF融合的主动配电网预测辅助状态估计(forecasting-aided state estimation, FASE)方法。首先,采用引力搜索算法(gravitational search algorithm, GSA)对支持向量回归(support vector regression, SVR)的关键参数进行优化处理,利用历史数据建立GSA-SVR模型,并将其引入至Att-LSTNet模型的输出层,构建一种增强预测模型。然后,利用UPF中的新息向量来训练该模型,并结合孤立森林算法和箱线图法对原始新息向量进行监控和修正。最后,针对量测噪声协方差矩阵未知的情况,结合修正后的新息向量和UPF计算出未知量测噪声协方差矩阵,并进行状态估计。基于IEEE33与IEEE118节点标准配电系统的算例结果表明,所提出的方法在估计精度、泛化能力和鲁棒性等方面具有优越性。     

离散卡尔曼滤波器滤波精度与噪声统计模型准确度的关系

针对多维渐近稳定线性离散系统,推导了使用铁初始条件和噪声统计的卡尔曼滤波器实际估计误差协方关矩阵,并基于该矩阵分析了滤波精度与模型精度的关系,指出使用非准确的初始条件和噪声统计模型 仍能构造出与最优卡尔曼滤器等效的滤波器,组合导航仿真系统的仿真验证了文中的结论。