Comments on “Performance evaluation of UKF-based nonlinear filtering”

This note points out that the main result of the paper [Xiong, K., Zhang, H. Y., & Chan, C. W. (2006). Performance evaluation of UKF-based nonlinear filtering. Automatica, 42(2), 261-270] can be extended to a class of so-called Gaussian filters. It justifies a practical countermeasure of the divergence, i.e., adding small quantities to the noise covariance matrix to stabilize the filter.     

Fault prediction of the nonlinear systems with uncertainty

Fault prediction which can forecast the fault in advance to avoid large calamity has attracted more and more attention. However, the current filter based fault prediction methods for the nonlinear systems are all based on the framework of the probability theory, and cannot realize fault prediction of the nonlinear systems with fuzzy uncertainty. Based on the extended fuzzy Kalman filter (EFKF) and the extended orthogonality principle, an improved fuzzy Kalman filter (IFKF) is firstly proposed to estimate the system states or the parameters in this paper. Then, according to the IFKF, a multi-step improved fuzzy Kalman predictor (MIFKP), which can be considered as an adaptive predictor, is obtained. Once the characteristic parameter is chosen, the MIFKP can be used to implement the multi-step fault prediction. Simulation results demonstrate that the proposed approach has the better prediction ability and stronger robustness than the traditional multi-step extended fuzzy Kalman predictor (MEFKP).     

H∞ nonlinear filtering

This paper investigates the problem of H_∞ estimation of nonlinear processes. An estimator, which may be nonlinear, is looked for so that a given bound on the ratio between the energy of the estimation error and the energy of the oxogeneous inputs to the estimated process is achieved. Conditions for the existence of such an estimator and formulas for its derivation are obtained using both the game theory approach and the theory of dissipative systems. The results of the paper extend the recent results on H_∞ nonlinear control. They are demonstrated by a simple example of a linear system with a nonlinear measurement rule and compared with corresponding results that are obtained by the extended Kalman filter.     

LPV model development and control of a solution copolymerization reactor

In this paper, linear parameter-varying (LPV) control is considered for a solution copolymerization reactor, which takes into account the time-varying nature of the parameters of the process. The nonlinear model of the process is first converted to an exact LPV model representation in the state-space form that has a large number of scheduling variables and hence is not appropriate for control design purposes due to the complexity of the LPV control synthesis problem. To reduce such complexity, two approaches are proposed in this paper. First, an approximate LPV representation with only one scheduling variable is obtained by means of a parameter set mapping (PSM). The second approach is based on reformulating the nonlinear model so that it provides an LPV model with a fewer number of scheduling parameters but preserves the same input–output behavior. Moreover, in the implementation of the LPV controllers synthesized with the derived models, the unmeasurable scheduling variables are estimated by an extended Kalman filter. Simulation results using the nonlinear model of the copolymerization reactor are provided in order to illustrate the performance of the proposed controllers in reducing the convergence time and the control effort.     

Noise covariance identification for time-varying and nonlinear systems

Kalman-based state estimators assume a priori knowledge of the covariance matrices of the process and observation noise. However, in most practical situations, noise statistics and initial conditions are often unknown and need to be estimated from measurement data. This paper presents an auto-covariance least-squares-based algorithm for noise and initial state error covariance estimation of large-scale linear time-varying (LTV) and nonlinear systems. Compared to existing auto-covariance least-squares based-algorithms, our method does not involve any approximations for LTV systems, has fewer parameters to determine and is more memory/computationally efficient for large-scale systems. For nonlinear systems, our algorithm uses full information estimation/moving horizon estimation instead of the extended Kalman filter, so that the stability and accuracy of noise covariance estimation for nonlinear systems can be guaranteed or improved, respectively.     

基于高斯—施密特粒子滤波器的多机器人协同定位

多机器人协同定位需对各个机器人的运动模型和观测模型精确建模,需要运用非线性、非高斯系统。已经应用于本领域的各种非线性算法主要有两种：一种是扩展卡尔曼滤波算法（EKF）,它对非线性系统进行局部线性化,从而间接利用卡尔曼算法进行滤波与估算;另一种是序列蒙特卡罗算法,即粒子滤波器（PF）。本文介绍了一种改进的粒子滤波
器,即高斯-施密特粒子滤波器（GHPF）,重点比较这三种算法在多机器人协同定位领域的应用效果。     

确定采样型强跟踪滤波飞机舵面故障诊断与隔离

为了克服扩展多模型自适应估计中扩展卡尔曼滤波的理论局限性,多重渐消因子强跟踪改进引起的滤波发散现象以及多维高斯故障概率计算量大等问题,本文将一类基于确定解析采样近似方法的非线性次优高斯滤波与多模型自适应估计相结合,提出了改进的多重渐消因子强跟踪非线性滤波快速故障诊断方法.确定采样型滤波克服了扩展卡尔曼滤波的理论局限性;推导了等效多重渐消因子计算方法,避免了非线性系统雅克比矩阵的计算,提高了故障突变时的跟踪性能;提出了基于平方根分解的改进的一步预测协方差更新方程,保证了滤波稳定性;提出了基于欧几里得范数简化的故障概率计算方法,降低了计算量.通过对比仿真验证了3种不同非线性滤波算法及其强跟踪改进算法的有效性,故障诊断方法跟踪性强、速度快、精度高,具有较好的鲁棒性和稳定性.     

Computing arrival cost parameters in moving horizon estimation using sampling based filters

Moving horizon estimation (MHE) is a numerical optimization based approach to state estimation, where the joint probability density function (pdf) of a finite state trajectory is sought, which is conditioned on a moving horizon of measurements. The joint conditional pdf depends on the a priori state pdf at the start of the horizon, which is a prediction pdf based on historical data outside the horizon. When the joint pdf is maximized, the arrival cost is a penalty term based on the a priori pdf in the MHE objective function. Traditionally, the a priori pdf is assumed as a multivariate Gaussian pdf and the extended Kalman filter (EKF) and smoother are used to recursively update the mean and covariance. However, transformation of moments through nonlinearity is poorly approximated by linearization, which can result in poor initialization of MHE. Sampling based nonlinear filters completely avoid Taylor series approximations of nonlinearities and attempt to approximate the non-Gaussian state pdf using samples and associated weights or probability mass points. The performance gains of sampling based filters over EKF motivate their use to formulate the arrival cost in MHE. The a priori mean and covariance are more effectively propagated through nonlinearities and the resulting arrival cost term can help to keep the horizon small. It is also possible to find closed-form approximations to the non-Gaussian a priori pdf from the sampling based filters. Thus, more realistic nonparametric arrival cost terms can be included by avoiding the Gaussian assumption. In this paper the use of the deterministic sampling based unscented Kalman filter, the class of random sampling based particle filter and the aggregate Markov chain based cell filter are discussed for initializing MHE. Two simulation examples are included to demonstrate the benefits of these methods over the traditional EKF approach.     

扩展风险性滤波算法的修正方法

针对非线性系统扩展风险性滤波(ERSF)估计精度较低的缺点,提出一种基于二阶Taylor展开对其进行高阶修正的算法.该算法利用高阶项对一阶扩展卡尔曼滤波(EKF)的状态估计向量及协方差阵进行适当修正,并由新息滤波方法得到修正的扩展风险性滤波(MERSF).高度非线性的仿真研究表明,所提出的算法在计算量增加不多的情况下,滤波精度有明显的提高.     

Online State–Space Modeling Using Recurrent Multilayer Perceptrons with Unscented Kalman Filter

A nonlinear black-box modeling approach using a state–space recurrent multilayer perceptron (RMLP) is considered in this paper. The unscented Kalman filter (UKF), which was proposed recently and is appropriate for state–space representation, is employed to train the RMLP. The UKF offers a derivative-free computation and an easy implementation, compared to the extended Kalman filter (EKF) widely used for training neural networks. In addition, the UKF has a fast convergence rate and an excellent capability of parameter estimation which are appropriate for online learning. Through modeling experiments of nonlinear systems, the effectiveness of the RMLP trained with the UKF is demonstrated.     

Online identification of nonlinear multivariable processes using self‐generating RBF neural networks

This paper addresses the problem of online model identification for multivariable processes with nonlinear and time‐varying dynamic characteristics. For this purpose, two online multivariable identification approaches with self‐organizing neural network model structures will be presented. The two adaptive radial basis function (RBF) neural networks are called as the growing and pruning radial basis function (GAP‐RBF) and minimal resource allocation network (MRAN). The resulting identification algorithms start without a predefined model structure and the dynamic model is generated autonomously using the sequential input‐output data pairs in real‐time applications. The extended Kalman filter (EKF) learning algorithm has been extended for both of the adaptive RBF‐based neural network approaches to estimate the free parameters of the identified multivariable model. The unscented Kalman filter (UKF) has been proposed as an alternative learning algorithm to enhance the accuracy and robustness of nonlinear multivariable processes in both the GAP‐RBF and MRAN based approaches. In addition, this paper intends to study comparatively the general applicability of the particle filter (PF)‐based approaches for the case of non‐Gaussian noisy environments. For this purpose, the Unscented Particle Filter (UPF) is employed to be used as alternative to the EKF and UKF for online parameter estimation of self‐generating RBF neural networks. The performance of the proposed online identification approaches is evaluated on a highly nonlinear time‐varying multivariable non‐isothermal continuous stirred tank reactor (CSTR) benchmark problem. Simulation results demonstrate the good performances of all identification approaches, especially the GAP‐RBF approach incorporated with the UKF and UPF learning algorithms. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

非线性动态滤波的迭代算法

阐述了标称状态的线性化方法和扩展的卡尔曼滤波公式及迭代卡尔曼滤波,探讨了非线性动态滤波的近似处理方法,围绕标称状态将非线性模型进行线性化,将标准的卡尔曼滤波扩展到非线性模型,得到扩展的卡尔曼滤波公式,研究了迭代滤波计算方法。扩展的卡尔曼滤波方法已经有效地用于非线性模型。     

A switch-mode information fusion filter based on ISRUKF for autonomous navigation of spacecraft

Aiming at the problem of loss of accuracy using extended Kalman filter (EKF) in case of orbit maneuver, this paper proposes a novel information fusion filtering algorithm-iterated square root unscented Kalman filter (ISRUKF), and then designs a switch-mode information fusion filter based on ISRUKF and extended Kalman filter (EKF). This method combines navigation sensors’ geocentric vector and geocentric distance with starlight angular distance, which efficiently improves the reliability of autonomous navigation. On this basis, the method deduced measurement function of information fusion. With a semi-physical simulation to verify the proposed method, the simulation results for stably running and orbital maneuvering spacecraft show that the switch-mode information fusion filter can reduce the complexity of the algorithm and ensure the accuracy of the estimation. Thus, the proposed switch-mode filter is very suitable for spacecraft autonomous navigation system and other strong nonlinear state estimation fields.     

Robust Kalman filtering for nonlinear multivariable stochastic systems in the presence of non‐Gaussian noise

The presence of outliers can considerably degrade the performance of linear recursive algorithms based on the assumptions that measurements have a Gaussian distribution. Namely, in measurements there are rare, inconsistent observations with the largest part of population of observations (outliers). Therefore, synthesis of robust algorithms is of primary interest. The Masreliez–Martin filter is used as a natural frame for realization of the state estimation algorithm of linear systems. Improvement of performances and practical values of the Masreliez‐Martin filter as well as the tendency to expand its application to nonlinear systems represent motives to design the modified extended Masreliez–Martin filter. The behaviour of the new approach to nonlinear filtering, in the case when measurements have non‐Gaussian distributions, is illustrated by intensive simulations. Copyright © 2015 John Wiley & Sons, Ltd.     

GNSS signals processing via linear and extended Kalman filters

In this paper, a new linear estimator that enables solution of the user position estimation problem has been designed for the case of global navigation satellite system (GNSS) utilization. The proposed recursive linear Kalman filter (LKF) does not need any linearization for nonlinear GNSS measurements and enables accurate estimation of the user coordinates and the distance measuring errors resulting from the time differences between the user and the satellite (clock bias). Comparison of the presented linear and conventional extended Kalman filters for the stationary and moving users' position estimation via GNSS measurements is performed. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Sonar-based robot navigation using nonlinear robust observers

This paper addresses the sonar-based navigation of mobile robots. The extended Kalman filtering (EKF) technique is considered, but from a deterministic, nonstochastic, point of view. For this problem, new results are presented on the robustness of the nonlinear observation scheme. The original feature is that the region-of-convergence question is posed in its complete nonlinear framework, that is, considering the dynamics not only of the estimation error ζ(t), but also of the covariance matrix P(t). In this way the approach followed makes less conservative the treatment and improves the convergence analysis. The proposed ideas were tested successfully on simulation experiments of a mobile platform.     

基于扩展卡尔曼粒子滤波算法的神经网络训练

神经网络的训练是一种非线性系统的辨识问题,基本粒子滤波算法已被成功用于训练神经网络,但基本粒子滤波算法在建议分布的选择上并没有考虑当前时刻观测值的影响,本文针对该问题提出使用扩展卡尔曼滤波器来生成建议分布。由于扩展卡尔曼滤波器在传递近似建议分布的均值和协方差的过程中充分利用了观测值信息,从而可以更好地描述神经网络权值的后验概率分布。实验结果证明,使用扩展卡尔曼滤波器作为建议分布的粒子滤波算法性能明显优于基本粒子滤波算法。     

含有多普勒频率的无迹卡尔曼滤波

针对扩展卡尔曼滤波不含多普勒频率和非线性的缺陷,提出一种新的含有多普勒频率的无迹卡尔曼滤波算法。新算法通过推导观测向量和状态向量之间的函数关系,结合无迹卡尔曼滤波思想,使滤波结果较扩展卡尔曼滤波算法具有更高的精度。最后对新老算法进行了仿真对比分析,结果表明了新算法的有效性和合理性。     

Conditions for stability of the extended Kalman filter and their application to the frequency tracking problem

The error dynamics of the extended Kalman filter (EKF), employed as an observer for a general nonlinear, stochastic discrete time system, are analyzed. Sufficient conditions for the boundedness of the errors of the EKF are determined. An expression for the bound on the errors is given in terms of the size of the nonlinearities of the system and the error covariance matrices used in the design of the EKF. The results are applied to the design of a stable EKF frequency tracker for a signal with time-varying frequency.This research was supported by the Co-operative Research Centre for Robust and Adaptive Systems ((CR)² ASys). The authors wish to acknowledge the funding of the activities of (CR)² ASys by the Australian Commonwealth Government under the Co-operative Research Centre Program.     

Unscented Kalman filter for continuous‐time nonlinear fractional‐order systems with process and measurement noises

This study proposes the design of unscented Kalman filter for a continuous‐time nonlinear fractional‐order system involving the process noise and the measurement noise. The nonlinear fractional‐order system is discretized to get the difference equation. According to the unscented transformation, the design method of unscented Kalman filter for a continuous‐time nonlinear fractional‐order system is provided. Compared with the extended Kalman filter, the proposed method can obtain a more accurate estimation effect. For fractional‐order systems containing non‐differentiable nonlinear functions, the method proposed in this paper is still effective. The unknown parameters are also discussed by the augmented vector method to achieve the state estimation and parameter identification. Finally, two examples are offered to verify the effectiveness of the proposed unscented Kalman filter for nonlinear fractional‐order systems.