含有分数阶有色关联噪声的分数阶系统的卡尔曼滤波器设计

提出基于Tustin生成函数的分数阶卡尔曼滤波器设计方法,以解决含有相互关联的分数阶有色过程噪声和分数阶有色测量噪声的连续时间线性分数阶系统的状态估计问题.通过Tustin生成函数方法,对连续时间线性分数阶系统进行离散化,将分数阶系统的微分方程转化为差分方程.利用增广向量法,将分数阶状态方程和分数阶有色噪声作为新的增广状态向量,从而将分数阶有色噪声转化为高斯白噪声.然后,提出一种基于Tustin生成函数的分数阶卡尔曼滤波算法,有效地实现对含有相互关联的分数阶有色过程噪声和分数阶有色测量噪声的连续时间线性分数阶系统的状态估计.与基于Grddotunwald-Letnikov差分的离散化方法相比,所提出的基于Tustin生成函数的卡尔曼滤波算法得到的状态估计精度更高,状态估计效果更好.最后,通过仿真结果验证所提出算法的有效性.     

Kalman Filters for Continuous-time Fractional-order Systems Involving Fractional-order Colored Noises Using Tustin Generating Function

This study presents fractional-order Kalman filers for linear fractional-order systems with colored noises using Tustin generating function. A continuous-time fractional-order system with the fractional-order colored process noise is discretized by Tustin generating function. The augmented vector consists of the state and the colored noise is offered to construct an augmented system based on the discretized state equation of a fractional-order system and the colored process noise. The Tustin fractional-order Kalman filter is designed based on the augmented system to obtain the state estimation, effectively. Besides, the colored noise involved in the measurement of a continuous-time fractional-order system is also discussed, and the corresponding Tustin fractional-order Kalman filter is provided in this study. Two illustrative examples are given to verify the effectiveness of Tustin fractional-order Kalman filters for the colored process and measurement noises.     

用于带有量测噪声系统的新型扩张状态观测器

提出了扩张状态观测器的一种新的形式,用于处理量测环节带有噪声干扰时的情况.ESO可以对不确定系统中的内外扰动进行观测,并以此为基础构成自抗扰控制器.但其性能会受量测噪声的影响.本文利用滤波器消除噪声的影响,并把已知的滤波器方程扩展到原有的ESO中,以补偿滤波器对实际输出信号的偏移作用.数字仿真表明,该方法可以有效的解决输出噪声对扩张状态观测器的影响.     

基于异质多传感器的网络分布数据融合的一种算法

针对多异质传感器数据融合能够实现信息互补,改善目标跟踪精度,提出了一种异质多传感器异步量测融合算法,即首先将量测方程线性化,再在砷合中心通过建立伪量测方程,得到同步的量测数据,然后利用噪声相关的伪序贯思想进行融合处理得到全局估计,与现有算法进行仿真比较,结果表明了该算法的有效性。     

Robust unscented Kalman filter with adaptation of process and measurement noise covariances

Unscented Kalman filter (UKF) has been extensively used for state estimation of nonlinear stochastic systems, which suffers from performance degradation and even divergence when the noise distribution used in the UKF and the truth in a real system are mismatched. For state estimation of nonlinear stochastic systems with non-Gaussian measurement noise, the Masreliez–Martin extended Kalman filter (EKF) gives better state estimates in relation to the standard EKF. However, the process noise and the measurement noise covariance matrices should be known, which is impractical in applications. This paper presents a robust Masreliez–Martin UKF which can provide reliable state estimates in the presence of both unknown process noise and measurement noise covariance matrices. Two numerical examples involving relative navigation of spacecrafts demonstrate that the proposed filter can provide improved state estimation performance over existing robust filtering approaches. Vision-aided robot arm tracking experiments are also provided to show the effectiveness of the proposed approach.     

Fractional-order Kalman filters for continuous-time linear and nonlinear fractional-order systems using Tustin generating function

This paper presents the fractional-order Kalman filters using Tustin generating function for linear and nonlinear fractional-order systems involving process noise and measurement noise. By using the Tustin generating function, the differential equation model is obtained by discretising the investigated continuous-time fractional-order system. The two kinds of fractional-order Kalman filters are given for the correlated and uncorrelated cases in terms of the process noise and measurement noise for linear fractional-order system, respectively. In addition, based on the first-order Taylor expansion formula, the extended fractional-order Kalman filter using Tustin generating function is proposed to improve the accuracy of state estimation. Finally, three examples are illustrated to verify the effectiveness of the Tustion fractional-order Kalman filters for linear and nonlinear fractional-order systems.     

A second-order stochastic filter involving coordinatetransformation

In the state estimation of a nonlinear system, the second-order filter is known to achieve better precision than the first-order filter [extended Kalman filter (EKF)] at the price of complex computation. If the measurement equation is linear in a transformed state variable, the complex measurement update equations of the second-order filter become as simple as the EKF case. Further, if the vector fields carrying the noise are constant, the high-order components in the variance propagation equation disappear. This suggests that if we make the measurement equation linear and make some vector fields constant through a coordinate transformation, we can simplify the second-order filter significantly while taking advantage of high precision. Finally, with an example of a falling body, we demonstrate through a Monte Carlo analysis the usefulness of the proposed method     

H∞次优滤波在角速度匹配传递对准中的应用

角速度匹配传递对准的量测量是主惯导的高精度陀螺测量信息与子惯导的陀螺测量信息之差,将角速度匹配传递对准的量测量送入卡尔曼滤波器进行信息融合,以获得导航系统误差的最优估计值.通过对角速度匹配传递对准的系统状态模型和量测模型的构建与分析,在将机翼弹性变形等干扰项为有色噪声时,采用H∞次优滤波的角速度匹配传递对准方式估计弹体的安装误差角与姿态失准角,从而实现机载导弹武器系统的快速传递对准.仿真结果表明,基于H∞次优滤波的角速度匹配传递对准,能有效抑制实际应用中弹体的机翼弹性变形等外界不确定噪声的干扰,在确保传递对准鲁棒性能的同时,可以获得良好的快速性能和滤波精度.因此,作为一种空中飞行的对准方法,采用H∞次优滤波将干扰项作为有色噪声应用于角速度匹配传递对准更符合工程应用的实际情况.     

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.     

Detection thresholds for tracking in clutter--A connection between estimation and signal processing

In the Kalman-Bucy filter and other trackers, the dependence of tracking performance upon the quality of the measurement data is well understood in terms of the measurement noise covariance matrix, which specifies the uncertainty in the values of the measurement inputs. The measurement noise and process noise covariances determine, via the Riccati equation, the state estimation error covariance. When the origin of the measurements is also uncertain, one has the widely studied problem of data association (or data correlation), and tracking performance depends critically on signal processing parameters, primarily the probabilities of detection and false alarm. In this paper we derive a modified Riccati equation that quantifies (approximately) the dependence of the state error covariance on these parameters. We also show how to use a receiver operating characteristic (ROC) curve in conjunction with the above relationship to determine the detection threshold in the signal processing system that provides measurements to the tracker so as to minimize tracking errors. The approach presented in this paper provides a feedback mechanism from the information processing (tracking) subsystem to the signal processing subsystem so as to optimize the overall performance in clutter.     

有色量测噪声下机器人同步定位与地图构建

针对有色量测噪声模型,提出一种有色量测噪声下的轮式机器人同步定位与地图构建算法。通过重新组合轮式机器人的过程模型和量测模型,将有色量测噪声量测模型转化为虚拟的白噪声量测模型。为使过程噪声和量测噪声不相关,对过程模型进行不相关条件处理。算法按照构造的虚拟过程模型和量测模型进行滤波估计和地图构建。仿真结果验证了算法的一致性和鲁棒性。     

具有噪声信息与状态模型不确定系统的IMM自适应滤波

卡尔曼滤波器广泛用于解决线性高斯系统的状态估计问题.然而,在实际应用中过程噪声和系统模型参数先验信息未知,且量测受到异常值干扰,给准确估计系统状态带来极大困难.针对具有噪声信息和状态模型不确定的动态系统,提出一种广义交互式多模型自适应滤波算法.该算法设计多个模型并行的方式对系统不确定进行处理,对于每个模型,建立Skew-T分布非对称重尾噪声表示模型,为了解决过程噪声与系统协方差相互耦合难以求解的问题,利用逆威沙特分布对系统预测协方差矩阵进行描述,并通过变分贝叶斯推理递归计算系统状态的后验分布.仿真结果和实验验证表明,在噪声信息和状态模型不确定条件下,所提出算法具有较高的估计精度.     

带有色厚尾量测噪声的鲁棒高斯近似滤波器和平滑器

为了解决带有色厚尾量测噪声的非线性状态估计问题,本文提出了新的鲁棒高斯近似（Gaussian approximate,GA）滤波器和平滑器.首先,基于状态扩展方法将量测差分后带一步延迟状态和白色厚尾量测噪声的非线性状态估计问题,转化成带厚尾量测噪声的标准非线性状态估计问题.其次,针对量测差分后模型中的噪声尺度矩阵和自由度（Degrees of freedom,DOF）参数未知问题,设计了新的高斯近似滤波器和平滑器,通过建立未知参数和待估计状态的共轭先验分布,并利用变分贝叶斯方法同时估计未知的状态、尺度矩阵、自由度参数.最后,利用目标跟踪仿真验证了本文提出的带有色厚尾量测噪声的鲁棒高斯近似滤波器和平滑器的有效性以及与现有方法相比的优越性.     

基于卡尔曼滤波算法的轨迹估计研究

在无线传感器网络中节点定位系统中,基于接收信号强度指示(RSSI)技术的定位算法研究有很多,这种定位技术成本低而且易于实现,但RSSI定位技术因容易受到环境因素的影响,在测距过程中,估测距离的误差很大。在RSSI定位系统的基础上,加入系统噪声和测量噪声,根据系统状态方程和动态系统测量方程,利用卡尔曼滤波算法,对RSSI进行滤波,并估测出移动节点的运动轨迹。仿真结果表明:改进卡尔曼滤波算法提高了移动节点的运动轨迹的定位精度。     

Adaptive Kalman filter for MEMS IMU data fusion using enhanced covariance scaling

MEMS (micro-electro-mechanical-system) IMU (inertial measurement unit) sensors are characteristically noisy and this presents a serious problem to their effective use. The Kalman filter assumes zero-mean Gaussian process and measurement noise variables, and then recursively computes optimal state estimates. However, establishing the exact noise statistics is a non-trivial task. Additionally, this noise often varies widely in operation. Addressing this challenge is the focus of adaptive Kalman filtering techniques. In the covariance scaling method, the process and measurement noise covariance matrices Q and R are uniformly scaled by a scalar-quantity attenuating window. This study proposes a new approach where individual elements of Q and R are scaled element-wise to ensure more granular adaptation of noise components and hence improve accuracy. In addition, the scaling is performed over a smoothly decreasing window to balance aggressiveness of response and stability in steady state. Experimental results show that the root mean square errors for both pith and roll axes are significantly reduced compared to the conventional noise adaptation method, albeit at a slightly higher computational cost. Specifically, the root mean square pitch errors are 1.1? under acceleration and 2.1? under rotation, which are significantly less than the corresponding errors of the adaptive complementary filter and conventional covariance scaling-based adaptive Kalman filter tested under the same conditions.     

一种带有色量测噪声的非线性系统辨识方法

利用最大似然判据, 本文提出了一种带有色量测噪声的非线性系统辨识方法. 首先, 利用量测差分方法将有色量测噪声白色化, 获得新的量测方程, 从而将带有色量测噪声的非线性系统辨识问题转化成带白色量测噪声和一步延迟状态的非线性系统辨识问题. 其次, 利用期望最大化(Expectation maximization, EM)算法提出了一种新的基于最大似然估计的非线性系统辨识方法, 该算法由期望步骤(Expectation step, E-step)和最大化步骤(Maximization step, M-step)两部分组成. 在期望步骤中, 基于当前估计的参数并利用带有色量测噪声的高斯近似滤波器和平滑器, 近似计算完整的对数似然函数的期望. 在最大化步骤中, 近似计算的似然函数期望值被最大化, 并且通过解析更新获得噪声参数估计, 通过Newton更新方法获得模型参数的估计. 最后, 数值仿真验证了本文提出算法的有效性.     

Dynamic data reconciliation: Alternative to Kalman filter

Process measurements are often corrupted with varying degrees of noise. Measurement noise undermines the performance of process monitoring and control systems. To reduce the impact of measurement noise, exponentially-weighted moving average and moving average filters are commonly used. These filters have good performance for processes under steady state or with slow dynamics. For processes with significant dynamics, more sophisticated filters, such as model-based filters, have to be used. The Kalman filter is a well known model-based filter that has been widely used in the aerospace industry. This paper discusses another model-based filter, the dynamic data reconciliation (DDR) filter. Both the Kalman and the DDR filters adhere to the same basic principle of using information from both measurements and models to provide a more reliable representation of the current state of the process. However, the DDR filter can more easily incorporated in a wide variety of model structures and is easier to understand and implement. Simulation results for a binary distillation column with four controlled variables showed that the DDR filters had equivalent performance to the Kalman filter in dealing with both white and autocorrelated noise.     

A treatment of nonwhite measurement noise in discrete linear systems

A combination of pure colored and pure white measurement noise may occur in several classes of linear discrete time dynamic stochastic systems of interest. Accurate state estimation in the presence of this nonwhite measurement noise is achieved through the use of a low order least squares filter. Previously the statistical nature of this nonwhite noise has been restricted, and computationally burdensome algorithms have been developed to handle this particular type of measurement noise. In overcoming these difficulties a filter is developed, similar in form to a Kalman filter, whose order is no higher than the order of the state.     

Sliding-mode estimators for a class of non-linear systems affected by bounded disturbances

The problem of state estimation for a class of non-linear systems with Lipschitz non-linearities is addressed using sliding-mode estimators. Stability conditions have been found to guarantee asymptotic convergence to zero of the estimation error in the absence of noise and non-divergence if the state perturbations and measurement noise are bounded. A method is proposed to find a suitable solution to the algebraic Riccati equation on which the design of the estimator is based. The performance of the resulting sliding-mode filter minimizes an upper bound on the asymptotic estimation error. Based on such an approach, a sliding-mode estimator may be designed so as to outperform the extended Kalman filter in practical applications with models affected by uncertainty and strong, possibly unknown non-linearities, as shown by means of simulations.     

Optimal filtering for polynomial system states with polynomial multiplicative noise

In this paper, the optimal filtering problem for polynomial system states with polynomial multiplicative noise over linear observations is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate and the error variance. As a result, the Ito differentials for the optimal estimate and error variance corresponding to the stated filtering problem are first derived. The procedure for obtaining a closed system of the filtering equations for any polynomial state with polynomial multiplicative noise over linear observations is then established, which yields the explicit closed form of the filtering equations in the particular cases of a linear state equation with linear multiplicative noise and a bilinear state equation with bilinear multiplicative noise. In the example, performance of the designed optimal filter is verified for a quadratic state with a quadratic multiplicative noise over linear observations against the optimal filter for a quadratic state with a state‐independent noise and a conventional extended Kalman–Bucy filter. Copyright © 2006 John Wiley & Sons, Ltd.