Distributed adaptive high‐gain extended Kalman filtering for nonlinear systems

In this work, we propose a distributed adaptive high‐gain extended Kalman filtering approach for nonlinear systems. Specifically, we consider a class of nonlinear systems that are composed of several subsystems interacting with each other via their states. In the proposed approach, an adaptive high‐gain extended Kalman filter is designed for each subsystem. The distributed Kalman filters communicate with each other to exchange estimated subsystem state information. First, assuming continuous communication among the distributed filters within deterministic form of subsystems, an implementation strategy that specifies how the distributed filters should communicate is designed and the detailed design of the subsystem filter is described. Second, we consider the case of stochastic subsystems for which the designed subsystem filters communicate to exchange information at discrete‐time instants. A state predictor in each subsystem filter is used to provide predictions of states of other subsystems. The stability properties of the proposed distributed estimation schemes with both continuous and discrete communications are analyzed. Finally, the effectiveness and applicability of the proposed schemes are illustrated via the application to a chemical process example. Copyright © 2017 John Wiley & Sons, Ltd.     

An iterated fuzzy extended Kalman filter for nonlinear systems

The fuzzy extended Kalman filter (FEKF) for state estimation can be used to deal with fuzzy uncertainty effectively. However, the linearisation processing of the FEKF introduces truncation error, which degrades the estimation precision. In order to reduce the error, a new iterated fuzzy extended Kalman filter (IFEKF), based on the FEKF and the maximum a posteriori estimation, is proposed in this article. Compared with the FEKF, the proposed algorithm can be used not only to deal with the fuzzy uncertainty, but also to reduce the truncation error and to estimate the states more accurately. With an algebraic example and a passive location simulation, it is shown that the IFEKF has better estimation precision than that of the FEKF.     

Event-triggered filtering and fault estimation for nonlinear systems with stochastic sensor saturations

This paper is concerned with the filtering problem for a class of nonlinear systems with stochastic sensor saturations and event-triggered measurement transmissions. An event-triggered transmission scheme is proposed with hope to ease the traffic burden and improve the energy efficiency. The measurements are subject to randomly occurring sensor saturations governed by Bernoulli-distributed sequences. Special effort is made to obtain an upper bound of the filtering error covariance in the presence of linearisation errors, stochastic sensor saturations as well as event-triggered transmissions. A filter is designed to minimise the obtained upper bound at each time step by solving two sets of Riccati-like matrix equations, and thus the recursive algorithm is suitable for online computation. Sufficient conditions are established under which the filtering error is exponentially bounded in mean square. The applicability of the presented method is demonstrated by dealing with the fault estimation problem. An illustrative example is exploited to show the effectiveness of the proposed algorithm.     

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.     

Distributed resilient estimation over sensor networks for nonlinear time‐delayed systems with stochastic perturbations

This paper is concerned with the distributed resilient estimation problem for a class of nonlinear time‐delayed systems subject to stochastic perturbations. The plant and the measurements are disturbed by two Gaussian white stochastic processes with known statistical information, respectively. In addition, a resilient estimator is designed for each node by means of the parameter uncertainties and Bernoulli‐distributed random variables. Then, a novel exponential‐bounded performance index is put forward to measure the disturbance rejection level of the distributed estimators against the external disturbances and the impact of the initial values. A new vector dissipation definition including multiple vectors of energy storage functions is established to deal with the time‐delay estimation error dynamics. Within the framework of local performance analysis inspired by this new definition of vector dissipation, sufficient conditions in terms of recursive linear matrix inequalities are constructed for each node to guarantee the desirable performance index. Next, a local optimization problem subject to a set of recursive linear matrix inequalities is presented for each node to minimize the upper bound in the performance index, where the calculations can be conducted on every node in a distributed manner and the estimator gains are also calculated. Finally, an illustrative simulation example is provided to verify the applicability of the proposed estimators.     

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.     

Robust Kalman filtering for uncertain discrete-time systems with probabilistic parameters bounded within a polytope

We consider the problem of finite horizon discrete-time Kalman filtering for systems with parametric uncertainties. Specifically, we consider unknown but deterministic uncertainties where the uncertain parameters are assumed to lie in a convex polyhedron with uniform probability density. The condition and a procedure for the construction of a suboptimal filter that minimizes an expected error covariance over-bound are derived.     

一种利用UKF的高速公路实时交通状态估计方法

把无轨迹卡尔曼滤波器（UKF）和宏观随机交通流模型结合在一起,可以实现对高速公路交通状态的实时估计。高速公路被看作是由等距离的路段首尾相接而成的系统,每个路段中交通变量的更新不光与其自身有关,还受到相邻路段的影响。交通传感器通常设置在路段的交界处,而且数量远少于所需估计的交通状态。采用压缩状态空间的形式,将模型参数也作为交通状态而非常量进行估计。仿真结果表明UKF方法能够有效地估计和跟踪交通状态的变化,并且与扩展卡尔曼滤波方法相比具有更高的精确度。     

Learning control for discrete-time nonlinear systems with sensor saturation and measurement noises

The iterative learning control (ILC) is investigated for a class of nonlinear systems with measurement noises where the output is subject to sensor saturation. An ILC algorithm is introduced based on the measured output information rather than the actual output signal. A decreasing sequence is also incorporated into the learning algorithm to ensure a stable convergence under stochastic noises. It is strictly proved with the help of the stochastic approximation technique that the input sequence converges to the desired input almost surely along the iteration axis. Illustrative simulations are exploited to verify the effectiveness of the proposed algorithm.     

Robust Kalman filtering for uncertain discrete‐time linear systems

This paper presents a steady‐state robust state estimator for a class of uncertain discrete‐time linear systems with norm‐bounded uncertainty. It is shown that if the system satisfies some particular structural conditions and if the uncertainty has a specific structure, the gain of the robust estimator (which assures a guaranteed cost) can be calculated using a formula only involving the original system matrices. Among the conditions the system has to satisfy, the strongest one relies on a minimum phase argument. It is also shown that under the assumptions considered, the robust estimator is in fact the Kalman filter for the nominal system. Copyright © 2003 John Wiley & Sons, Ltd.     

基于Kalman滤波的带相关噪声系统统一的Wiener状态估值器

基于Kalman滤波和白噪声估值器, 对带非零均值相关噪声系统提出了渐近稳定的统一的和通用的Wiener状态估值器. 它们可统一处理滤波、平滑和预报问题, 且避免了计算最优初始状态估值. 它们揭示了Kalman滤波器和Wiener滤波器之间的关系.一个仿真例子说明其有效性.     

无线传感器网络环境下基于卡尔曼滤波的PTP协议

作为分布式系统的重要组成部分,精确时间同步是对时间敏感的工业无线网络的核心技术.基于时间信息包交换的IEEE 1588精确时间同步协议（PTP）主要针对有线网络提出,其同步精度受制于时间戳的精度和传输延迟抖动.在无线传感网中,节点难以获取精确时钟戳,同时由于信道共享、包冲突和信道衰落,无线网络的传输延迟抖动非常明显. 研究了无线网络中PTP的性能与时间戳精度之间的关系,提出了一个自回归模型来描述时钟漂移,将PTP中的包交换过程抽象为一组状态空间方程,将延迟抖动等作为观测噪音,从而利用卡尔曼滤波器予以滤除.仿真结果表明,在不同时间戳精度和延迟抖动下,卡尔曼滤波能有效改善时钟误差和稳定性.     

A novel strong tracking finite-difference extended Kalman filter for nonlinear eye tracking

Non-intrusive methods for eye tracking are important for many applications of vision-based human computer interaction. However, due to the high nonlinearity of eye motion, how to ensure the robustness of external interference and accuracy of eye tracking poses the primary obstacle to the integration of eye movements into todays’s interfaces. In this paper, we present a strong tracking finite-difference extended Kalman filter algorithm, aiming to overcome the difficulty in modeling nonlinear eye tracking. In filtering calculation, strong tracking factor is introduced to modify a priori covariance matrix and improve the accuracy of the filter. The filter uses finite-difference method to calculate partial derivatives of nonlinear functions for eye tracking. The latest experimental results show the validity of our method for eye tracking under realistic conditions. Supported by the National Natural Science Foundation of China (Grant No. 60572027), the Outstanding Young Researchers Foundation of Sichuan Province (Grant No. 03ZQ026-033), the Program for New Century Excellent Talents in University of China (Grant No. NCET-05-0794), and the Young Teacher Foundation of Mechanical School (Grant No. MYF0806)     

WSN中一种基于扩展卡尔曼滤波器的虚假数据注入检测算法

为了有效地检测传感器网络中被注入的虚假数据,提出一种基于扩展卡尔曼滤波器（EKF）的虚假数据注入检测算法。首先通过监控邻近节点行为,使用EKF预测邻近节点未来状态;然后给出了使用不同的融合函数（平均、求和、最大、最小）时理论阈值的确定方法;最后为了克服本地检测机制的缺陷,将本地检测方法与系统监控模块有效配合,从而准确地区分出恶性事件和紧急事件。仿真实验结果表明,无论是在合成数据还是实时数据下进行测试,该算法都能为无线传感器网络进行安全的数据融合提供有效的入侵检测功能。     

非线性随机离散系统推广卡尔曼滤波方法收敛性分析

讨论了非线性随机离散系统的推广卡尔曼滤波算法的收敛性 .基于BoutayebM的一阶线性化技巧 ,得到了确保局部渐近收敛的充分条件     

带扩展卡尔曼滤波的柔性关节机器人虚拟分解控制

针对柔性关节机器人在非完全状态反馈条件下的轨迹跟踪控制问题,本文提出一种基于虚拟分解控制(virtual decomposition control,VDC)理论和扩展卡尔曼滤波(extended Kalman filtering,EKF)观测的控制方法.首先,考虑模型参数的不确定性和外界扰动因素,分别设计刚性连杆子系统和柔性关节子系统的虚拟分解控制律.然后,为突破现有VDC方法依赖于全状态反馈测量的局限,设计一种基于EKF的间接状态观测器,实现了仅需电机侧位置和速度测量而不需连杆侧任何状态信息测量的闭环控制.此外,结合虚拟稳定和李雅普诺夫稳定理论给出了严格的系统稳定性证明.最后,实例对比仿真验证了所提出控制算法的有效性,且相比于基于传统拉格朗日整体动力学的典型算法,具有更优的轨迹跟踪性能.     

Optimal guaranteed cost filtering for uncertain discrete-time linear systems

This paper presents a result on the design of a steady-state robust state estimator for a class of uncertain discrete-time linear systems with normal bounded uncertainty. This result extends the steady state Kalman filter to the case in which the underlying system is uncertain. A procedure is given for the construction of a state estimator which minimizes a bound on the state error covariance. It is shown that this leads to a state estimator which is optimal with respect to a notion of quadratic guaranteed cost state estimation.     

Observer‐based controller for constrained uncertain stochastic nonlinear discrete‐time systems

In this paper, an observer‐based control approach is proposed for uncertain stochastic nonlinear discrete‐time systems with input constraints. The widely used extended Kalman filter (EKF) is well known to be inadequate for estimating the states of uncertain nonlinear dynamical systems with strong nonlinearities especially if the time horizon of the estimation process is relatively long. Instead, a modified version of the EKF with improved stability and robustness is proposed for estimating the states of such systems. A constrained observer‐based controller is then developed using the state‐dependent Riccati equation approach. Rigorous analysis of the stability of the developed stochastically controlled system is presented. The developed approach is applied to control the performance of a synchronous generator connected to an infinite bus and chaos in permanent magnet synchronous motor. Simulation results of the synchronous generator show that the estimated states resulting from the proposed estimator are stable, whereas those resulting from the EKF diverge. Moreover, satisfactory performance is achieved by applying the developed observer‐based control strategy on the two practical problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Discrete‐time invariant extended Kalman filter on matrix Lie groups

In this article, we derive symmetry preserving discrete‐time invariant extended Kalman filters (IEKF) on matrix Lie groups. These Kalman filters offer an advantage over classical extended Kalman filters as the error dynamics for such filters are independent of the group configuration which, in turn, provides a uniform estimate of the region of convergence. In contrast to existing techniques in the literature, the discrete‐time IEKF is derived using minimal tools from differential geometry which simplifies the derivation and the representation of IEKF. In our technique, the linearized error dynamics is defined on the Lie algebra directly using variational approaches, unlike conventional approaches where the error dynamics is translated to an Euclidean space using the logarithm map before its linearization. Moreover, the Kalman gains and its associated difference Riccati equations are derived in operator spaces by setting a discrete‐time optimal control problem and solving it with discrete‐time Pontryagin's maximum principle. The proposed discrete‐time IEKF is implemented for the attitude dynamics of the rigid body, which is a benchmark problem in control. It is observed from the numerical studies that the IEKF is computationally less intensive and provides better performance than the classical extended Kalman filter.