Distributed fusion filter for multi-sensor systems with finite-step correlated noises

This paper addresses the distributed fusion filtering problem for multi-sensor systems with finite-step correlated noises. The process noise and observation noises of different sensors are finite-step auto- and cross-correlated, respectively. Based on the optimal local filtering algorithms that we presented before, the filtering error cross-covariance matrices between any two local filters are derived based on an innovation analysis approach. A distributed fusion filter is put forward by using matrix-weighted fusion estimation algorithm in the linear unbiased minimum variance sense. Finally, the proposed algorithms are extended to systems with random parameter matrices. Two simulation examples are given to show the effectiveness of the proposed algorithms.     

Distributed weighted robust Kalman filter fusion for uncertain systems with autocorrelated and cross-correlated noises

In this paper, the problem of distributed weighted robust Kalman filter fusion is studied for a class of uncertain systems with autocorrelated and cross-correlated noises. The system under consideration is subject to stochastic uncertainties or multiplicative noises. The process noise is assumed to be one-step autocorrelated. For each subsystem, the measurement noise is one-step autocorrelated, and the process noise and the measurement noise are two-step cross-correlated. An optimal robust Kalman-type recursive filter is first designed for each subsystem. Then, based on the newly obtained optimal robust Kalman-type recursive filter, a distributed weighted robust Kalman filter fusion algorithm is derived for uncertain systems with multiple sensors. The distributed fusion algorithm involves a recursive computation of the filtering error cross-covariance matrix between any two subsystems. Compared with the centralized Kalman filter, the distributed weighted robust Kalman filter developed in this paper has stronger fault-tolerance ability. Simulation results are provided to demonstrate the effectiveness of the proposed approaches.     

Optimal linear filter design for systems with correlation in the measurement matrices and noises: recursive algorithm and applications

This paper addresses the optimal least-squares linear estimation problem for a class of discrete-time stochastic systems with random parameter matrices and correlated additive noises. The system presents the following main features: (1) one-step correlated and cross-correlated random parameter matrices in the observation equation are assumed; (2) the process and measurement noises are one-step autocorrelated and two-step cross-correlated. Using an innovation approach and these correlation assumptions, a recursive algorithm with a simple computational procedure is derived for the optimal linear filter. As a significant application of the proposed results, the optimal recursive filtering problem in multi-sensor systems with missing measurements and random delays can be addressed. Numerical simulation examples are used to demonstrate the feasibility of the proposed filtering algorithm, which is also compared with other filters that have been proposed.     

Optimal state estimation for networked systems with random parameter matrices,correlated noises and delayed measurements

In this paper, the optimal least-squares state estimation problem is addressed for a class of discrete-time multisensor linear stochastic systems with state transition and measurement random parameter matrices and correlated noises. It is assumed that at any sampling time, as a consequence of possible failures during the transmission process, one-step delays with different delay characteristics may occur randomly in the received measurements. The random delay phenomenon is modelled by using a different sequence of Bernoulli random variables in each sensor. The process noise and all the sensor measurement noises are one-step autocorrelated and different sensor noises are one-step cross-correlated. Also, the process noise and each sensor measurement noise are two-step cross-correlated. Based on the proposed model and using an innovation approach, the optimal linear filter is designed by a recursive algorithm which is very simple computationally and suitable for online applications. A numerical simulation is exploited to illustrate the feasibility of the proposed filtering algorithm.     

Optimal Fusion Reduced-Order Kalman Filters Weighted by Scalars for Stochastic Singular Systems

Based on the optimal fusion algorithm weighted by scalars in the linear minimum variance sense, a distributed optimal fusion reduced-order Kalman filter with scalar weights is presented for discrete-time stochastic singular systems with multiple sensors and correlated noises. It has higher accuracy than any local filter does. Compared with the distributed fusion filter weighted by matrices, it has lower accuracy but has reduced computational burden. Computation formula of cross-covariance matrix of the filtering errors between any two sensors is given. An example with three sensors shows the effectiveness.     

Distributed fusion estimation for multi-sensor asynchronous sampling systems with correlated noises

This paper is concerned with the distributed fusion estimation problem for a class of multi-sensor asynchronous sampling systems with correlated noises. The state updates uniformly and the sensors sample randomly. Based on the measurement augmentation method, the asynchronous sampling system is transformed to the synchronous sampling one. Local filter is designed by using an innovation analysis approach. Then, the filtering error cross-covariance matrix between any two local filters is derived. Finally, the optimal distributed fusion filter is proposed by using matrix-weighted fusion algorithm in the linear minimum variance sense. Simulation results show the effectiveness of the proposed algorithms.     

Multi-sensor information fusion white noise filter weighted by scalars based on Kalman predictor

A unified multi-sensor optimal information fusion criterion weighted by scalars is presented in the linear minimum variance sense. The criterion considers the correlation among local estimation errors, only requires the computation of scalar weights, and avoids the computation of matrix weights so that the computational burden can obviously be reduced. Based on this fusion criterion and Kalman predictor, an optimal information fusion filter for the input white noise, which can be applied to seismic data processing in oil exploration, is given for discrete time-varying linear stochastic control systems measured by multiple sensors with correlated noises. It has a two-layer fusion structure. The first fusion layer has a netted parallel structure to determine the first-step prediction error cross-covariance for the state and the filtering error cross-covariance for the input white noise between any two sensors at each time step. The second fusion layer is the fusion center to determine the optimal scalar weights and obtain the optimal fusion filter for the input white noise. Two simulation examples for Bernoulli-Gaussian white noise filter show the effectiveness.     

Distributed optimal fusion steady-state Kalman filter for systems with coloured measurement noises

Based on the optimal fusion criterion weighted by matrices in the linear minimum variance sense, an optimal information fusion steady-state Kalman filter is given for the discrete time-invariant linear stochastic control system measured by multiple sensors with coloured measurement noises, which is equivalent to an optimal information fusion steady-state Kalman predictor with a two-layer fusion structure for system with correlated noises. Furthermore, the steady-state optimal fusion predictor can be obtained only by fusing once after all local subsystems enter the steady-state predictions. The solution of steady-state prediction error cross-covariance matrix between any two subsystems can be obtained by iteration with an arbitratry initial value, whose convergence is proved. Applying it to a tracking system with three sensors shows its effectiveness.     

Optimal distributed Kalman filtering fusion for a linear dynamic system with cross-correlated noises

In this article, we study the distributed Kalman filtering fusion problem for a linear dynamic system with multiple sensors and cross-correlated noises. For the assumed linear dynamic system, based on the newly constructed measurements whose measurement noises are uncorrelated, we derive a distributed Kalman filtering fusion algorithm without feedback, and prove that it is an optimal distributed Kalman filtering fusion algorithm. Then, for the same linear dynamic system, also based on the newly constructed measurements, a distributed Kalman filtering fusion algorithm with feedback is proposed. A rigorous performance analysis is dedicated to the distributed fusion algorithm with feedback, which shows that the distributed fusion algorithm with feedback is also an optimal distributed Kalman filtering fusion algorithm; the P matrices are still the estimate error covariance matrices for local filters; the feedback does reduce the estimate error covariance of each local filter. Simulation results are provided to demonstrate the validity of the newly proposed fusion algorithms and the performance analysis.     

A new approach to distributed fusion filtering for networked systems with random parameter matrices and correlated noises

This paper is concerned with the distributed filtering problem for a class of discrete-time stochastic systems over a sensor network with a given topology. The system presents the following main features: (i) random parameter matrices in both the state and observation equations are considered; and (ii) the process and measurement noises are one-step autocorrelated and two-step cross-correlated. The state estimation is performed in two stages. At the first stage, through an innovation approach, intermediate distributed least-squares linear filtering estimators are obtained at each sensor node by processing available output measurements not only from the sensor itself but also from its neighboring sensors according to the network topology. At the second stage, noting that at each sampling time not only the measurement but also an intermediate estimator is available at each sensor, attention is focused on the design of distributed filtering estimators as the least-squares matrix-weighted linear combination of the intermediate estimators within its neighborhood. The accuracy of both intermediate and distributed estimators, which is measured by the error covariance matrices, is examined by a numerical simulation example where a four-sensor network is considered. The example illustrates the applicability of the proposed results to a linear networked system with state-dependent multiplicative noise and different network-induced stochastic uncertainties in the measurements; more specifically, sensor gain degradation, missing measurements and multiplicative observation noises are considered as particular cases of the proposed observation model.     

广义系统信息融合稳态与自校正满阶Kalman滤波器

基于线性最小方差标量加权融合算法和射影理论,对带多个传感器和带相关噪声的广义系统,提出了分布式标量加权融合稳态满阶Kalman滤波器.推得了任两个传感器子系统之间的稳态满阶滤波误差互协方差阵,其解可任选初值离线迭代计算.所提出的稳态融合滤波器避免了每时刻计算协方差阵和融合权重,减小了在线计算负担.当系统含有未知模型参数时,基于递推增广最小二乘算法和标量加权融合算法,提出了一种两段融合自校正状态滤波器.其中第1段融合获得未知参数的融合估计;第2段融合获得分布式自校正融合状态滤波器.与局部估计和加权平均融合估计相比,所提出的标量加权融合参数估计和自校正状态估计都具有更高的精度.仿真研究验证了其有效性.     

Distributed optimal component fusion weighted by scalars for fixed-lag Kalman smoother

Based on the optimal fusion criterion in the linear minimum variance sense, a distributed optimal fusion fixed-lag Kalman smoother with a three-layer fusion structure is given for the discrete time-varying linear stochastic control systems with multiple sensors and correlated noises. Its components are estimated by scalar weighting fusion, respectively. It only requires in parallel a series of computations of the weighted scalars, and avoids the computations of the weighted matrices, so that the computational burden can obviously be reduced. Further, the steady-state fusion smoother is also given for the discrete time-invariant linear stochastic control systems. The scalar weights can be obtained by fusing once after all local estimations reach steady state. It can reduce the online computational burden. Also, the computation formulas of smoothing error cross-covariance matrices are given. Two simulation examples show the performance.     

带相关噪声异步采样系统的分布式最优线性递推融合估计

对带相关噪声的异步均匀采样线性离散系统, 研究了分布式最优线性递推融合预报和滤波问题. 通过引入 满足伯努利分布的随机变量将系统同步化, 给出了局部Kalman预报器和滤波器. 分别推导了局部估值间的互协方 差阵、分布式最优线性融合估值与局部估值间的互协方差阵. 提出了分布式最优线性递推融合预报器和滤波器. 与 局部估值按矩阵加权的分布式融合估计算法相比, 所提出的算法具有更高的估计精度, 但与集中式融合相比有精度 损失. 为了进一步提高估计精度, 又提出了带反馈的分布式最优线性递推融合预报器和滤波器, 证明了带反馈的融 合估计与集中式融合估计具有相同的精度. 仿真例子验证了所提算法的有效性.     

Multi-sensor optimal fusion fixed-interval Kalman smoothers

Based on the optimal weighted fusion algorithms in the linear minimum variance sense, the optimal fusion fixed-interval Kalman smoothers are given for discrete time-varying linear stochastic control systems with multiple sensors and correlated noises, which have a three-layer fusion structure. The first and the second fusion layers both have netted parallel structures to determine the cross-covariance matrices of prediction and smoothing errors between any two-sensor subsystems, respectively. The third fusion layer is the fusion centre to determine the optimal weights and obtain the optimal fusion fixed-interval smoothers. Smoothing error cross-covariance matrix between any two-sensor subsystems is derived. Applying it to a tracking system with three-sensors shows the effectiveness.     

带有色观测噪声的广义系统Kalman滤波器

对于带自回归滑动平均(ARMA)有色观测噪声的多传感器为广义离散随机线性系统,应用奇异值分解,将其变换为等价的两个降阶多传感器子系统,提出了广义系统多传感器信息融合状态滤波问题。为了提高精度,采用Kalman滤波方法,在线性最小方差按块对角阵最优加权融合准则下,给出了按矩阵加权解耦的分布式Kalman滤波器,可减少计算负担和改善局部滤波精度。为了计算最优加权,提出了局部滤波误差协方差阵的计算公式。一个Monte Carlo仿真例子说明了方法的有效性。     

Minimum variance estimation for linear uncertain systems with one-step correlated noises and incomplete measurements

This paper deals with state estimation problem for linear uncertain systems with correlated noises and incomplete measurements. Multiplicative noises enter into state and measurement equations to account for the stochastic uncertainties. And one-step autocorrelated and cross-correlated process noises and measurement noises are taken into consideration. Using the latest received measurement to compensate lost packets, the modified multi-step random delays and packet dropout model is adopted in the present paper. By augmenting system states, measurements and new defined variables, the original system is transformed into the stochastic parameter one. On this basis, the optimal linear estimators in the minimum variance sense are designed via projection theory. They depend on the variances of multiplicative noises, the one-step correlation coefficient matrices together with the probabilities of delays and packet losses. The sufficient condition on the existence of steady-state estimators is then given. Finally, simulation results illustrate the performance of the developed algorithms.     

Unscented Kalman Filtering for Nonlinear State Estimation with Correlated Noises and Missing Measurements

The unscented Kalman filtering problem is investigated for a class of nonlinear discrete stochastic systems subject to correlated noises and missing measurements. Here, a random variable obeying Bernoulli distribution with known conditional probability is introduced to depict the phenomenon of missing measurements occurring in a stochastic way. Due to taking the correlation of noises into account, a one-step predictor is designed by applying the innovative analysis and unscented transformation approach. And then, based on one-step predictor and the minimum mean square error principle, a new unscented Kalman filtering algorithm is proposed such that, for the correlated noises and missing measurements, the filtering error is minimized. By solving the recursive matrix equation, the filter gain matrices and the error covariance matrices can be obtained and the proposed results can be easily verified by using the standard numerical software. We finally provide a numerical example to show the performance of the proposed approach.     

Optimal sequential and distributed fusion for state estimation in cross-correlated noise

This paper is concerned with the optimal state estimation for linear systems when the noises of different sensors are cross-correlated and also coupled with the system noise of the previous step. We derive the optimal linear estimation in a sequential form and for distributed fusion. They are both compared with the optimal batch fusion, suboptimal batch fusion, suboptimal sequential fusion, and the suboptimal distributed fusion where the cross-correlation between the noises are neglected. The comparison is in terms of theoretical filter mean square error and the real root mean square error. Simulation on a target tracking example is given to show the effectiveness of the presented algorithms.     

带未知通信干扰和丢包补偿的多传感器网络化不确定系统的分布式融合滤波

研究了带有未知通信干扰、观测丢失和乘性噪声不确定性的多传感器网络化系统的状态估计问题.通过白色乘性噪声描述系统状态和观测中的随机不确定性,采用一组服从Bernoulli分布的随机变量描述网络传输过程中存在的观测丢失现象,且数据传输中存在未知的网络通信干扰.当发生丢包时,以当前丢失观测的预报值进行补偿.对每个单传感器子系统,应用线性无偏最小方差估计准则设计了不依赖于未知通信干扰的最优线性滤波器.推导了任两个局部滤波误差之间的互协方差阵.进而,应用矩阵加权融合估计算法给出了分布式融合状态滤波器.仿真例子验证了算法的有效性.     

Multisensor optimal information fusion input white noise deconvolution estimators

The unified multisensor optimal information fusion criterion weighted by matrices is rederived in the linear minimum variance sense, where the assumption of normal distribution is avoided. Based on this fusion criterion, the optimal information fusion input white noise deconvolution estimators are presented for discrete time-varying linear stochastic control system with multiple sensors and correlated noises, which can be applied to seismic data processing in oil exploration. A three-layer fusion structure with fault tolerant property and reliability is given. The first fusion layer and the second fusion layer both have netted parallel structures to determine the first-step prediction error cross-covariance for the state and the estimation error cross-covariance for the input white noise between any two sensors at each time step, respectively. The third fusion layer is the fusion center to determine the optimal matrix weights and obtain the optimal fusion input white noise estimators. The simulation results for Bernoulli-Gaussian input white noise deconvolution estimators show the effectiveness.