Fusion of Multiple Estimates by Covariance Intersection: Why and Howit Is Suboptimal

In this paper, the problem of fusion estimation for networked multi-sensor systems with multiplicative noise and correlated noise is studied. First, the state space model is transformed into a new system with fictitious noises and local Kalman filter are obtained. Then the sequential inverse covariance intersection (SICI) fusion algorithm is applied and the SICI fusion estimator is proposed to avoid the computation burden of local estimators. Compared with the Sequential Covariance Intersection (SCI) fusion algorithm, SICI fusion algorithm is less conservative, and its estimation accuracy is higher than that of local filter and SCI fusion estimator. The validity and consistency of the proposed fusion estimator are verified by a simulation example.

In this note, a hierarchical fusion estimation method is presented for clustered sensor networks with a very general setup where sensors (sensor nodes) and estimators (cluster heads) are allowed to work asynchronously with aperiodic sampling and estimation rates. A sequential measurement fusion (SMF) method is presented to design local estimators, and it is shown that the SMF estimator is equivalent to the measurement augmentation (MA) estimator in precision but with much lower computational complexity. Two types of sequential covariance intersection (CI) fusion estimators are presented for the fusion estimation. The proposed SCI fusion estimators provide a satisfactory estimation precision that is close to the centralized batch CI (BCI) estimator while requiring smaller computational burden as compared with the BCI estimator. Therefore, the proposed hierarchical fusion estimation method is suitable for real-time applications in asynchronous sensor networks with energy constraints. Moreover, the method is applicable to the case with packet delays and losses.

In several practical applications, an economical processing of multi-sensor data is realised by the fusion of estimates. A very cheap solution, which is also very conservative, is the Covariance Intersection (CI) fusion. Basic information configurations of the estimation system have been inspected by the authors in a preceding paper [1]. This paper provides an extension of the previous discussion to the fusion with memory. It shows that the adding of memory cannot improve the fusions without memory given by partial or full information feedback. Conversely, it shows that the fusion without memory and no information feedback can be in some cases improved by the adding of memory.

For multi-sensor system with colored measurement noises, using the observation transformation, the system can be converted into an equivalent system with correlated measurement noises. Based on this method, using the classical Kalman filtering, this paper proposed a covariance intersection (CI) fusion Kalman estimator, which can handle the fused filtering, prediction and smoothing problems. The advantage of the proposed method is that it can avoid the computation of the cross co-variances among the local filtering errors and can reduce the computational burden significantly, as well as the CI fusion algorithm can be used in the uncertain system with unknown cross covariances. Based on classical Kalman filtering theory, the centralized fusion and three weighted fusion (weighted by matrices, scalars and diagonal) estimators are also presented respectively. Their accuracy comparisons are given. The geometric interpretations based on covariance ellipses are also given. A Monte-Carlo simulation example shows that the accuracy relations.

A recently proposed algorithm of fusion under partially known correlations of estimation errors has been proved to outperform the classic Covariance Intersection algorithm, which was proposed for the case of no knowledge of correlation. This paper shows that the assumptions of the recently proposed algorithm are rather strict with respect to the classic one. Namely, the mean square error (MSE) matrices of the two state estimates cannot be upper-bounded arbitrarily. A relaxation of the assumption that the matrices have to be known exactly is discussed, as well as an iterative fusion of multiple estimates, and several examples dealing with dependent errors are provided.

In a distributed estimation system, the local sensor estimates are transferred to fusion center and fused to be an optimal estimation according to some criterion. The well-known best linear unbiased estimation (BLUE) fusion optimally combines the local estimates by minimising the covariance matrix of estimation errors. However, the BLUE fusion need know the cross-covariances of sensor estimation errors exactly, which are not attainable in most of practical applications. In this article, the random matrix is employed to describe the uncertainty of the estimation error covariance among the sensors. By minimising the estimation error covariance only for the most favorable realisations of the random matrix, we model it as an optimisation problem with chance constraint. With appropriate relaxation method, a robust linear unbiased estimation fusion is proposed analytically. Furthermore, an upper bound on the relaxation gap is proposed. Finally, we demonstrate through some numerical simulations that the presented estimation fusion always has smaller absolute estimation error than that of the covariance intersection filter.

Distributed diffusion unscented Kalman filtering based on covariance intersection with intermittent measurements

Covariance Intersection fusion is bound-optimal under unknown correlations. Partial knowledge can improve the fusion. An implicit constraint on correlation has been introduced in the literature for full-vector estimates. This paper considers fusion of partial estimates. Weak and strong counterparts of the full-vector assumption are proposed. An analysis of admissible ideal fusions reveals that the considered implicit partial knowledge cannot improve the Covariance Intersection fusion of partial estimates. An exception is found for the strong assumption and fusion of one partial and one full-vector estimate. For this case, an improved fusion rule is presented.

We extend our obtained fusion formula yielding an optimal mean square combination of unbiased estimates on the case of an arbitrary number of biased estimates and apply it to fusion of multisensor local estimates. The suboptimal two-stage filter for linear dynamic systems is designed: the local optimal Kalman estimates computed at the first stage are linearly fused at the second stage. Its realization needs a lower memory demand than the optimal Kalman filter. The example exhibits the effect of noise on the performance of fusion of the state estimates based on measurements from different sensors.

SummaryFor networked mixed uncertain time‐varying systems with uncertain noise variances, random one‐step measurement delay, state‐dependent and noise‐dependent multiplicative noises, and linearly dependent additive white noises, the robust local, centralized, and distributed fusion estimation problems are addressed. Three new approaches are presented, which include a new augmented state approach with fictitious white noises, an extended Lyapunov equation approach with two Lyapunov equations, and a universal integrated covariance intersection (ICI) fusion approach of integrating the minimax robust local Kalman estimators and their conservative cross‐covariances. They constitute a new important methodology of solving robust fusion estimation problems. Applying them, the local, centralized, and distributed ICI fusion time‐varying and steady‐state robust Kalman estimators (predictor, filter, and smoother) are presented in the sense that for all admissible uncertainties, their actual estimation error variances are guaranteed to have the corresponding minimal upper bounds. Their robustness, convergence, and accuracy relations are proved. Specially, the proposed ICI fusers improve the robust accuracies of the original covariance intersection fusers, and overcome their drawbacks, such that the local estimators and their conservative variances are assumed to be known, and the conservative cross‐variances are ignored. A simulation example with application to a vehicle suspension system shows the effectiveness of the proposed approaches and results.

For the practical multisensor systems, the communication channels are often unreliable, especially in large, wireless, multi hop sensor networks. Thus, the communication delays often appear and produce random and unknown influence upon the performance of fusion estimators. Recently, data fusion for "Out-of-sequence" measurements (OOSM) attracts a mass of attentions gradually and some filters based on single sensor and multiple sensors have been reported. However, the existing methods for OOSM problem have several shortcomings, such as limited application for single sensor, bad real-time performance for the smooth property, low fusion estimate for neglecting the delayed information, and so forth. To tackle these problems, a novel batch fusion estimator based on the relative measurements for multisensor tracking systems with OOSM is proposed. The estimator is optimal in linear minimum mean square error (LMMSE). The paper also presents the suboptimal recursive form of the proposed estimator. Its principle is to establish the relative measurement equations of delayed observations to the next latest systemic state. Compared with some existingfusion methods for OOSM, the proposed method with arbitrary time delays has the best total performance. At the same time, algorithm analysis and computer simulation validate the advantages of the proposed fusion estimator.

This article is concerned with the nonlinear state estimation for the multisenor networked system with uncertain bounded noise. Traditional distributed methods only pay attention to the information fusion of state estimations, but neglect the fusion of noise statistics. The difference of noise statistics among sensor nodes usually affects the precision of state estimation in wireless sensor networks, especially for the distributed state estimation. In this article, in order to improve the accuracy of noise statistic estimations, a distributed noise statistic estimator is derived based on covariance intersection criterion and modified Sage–Husa maximum posterior. Then, distributed adaptive cubature information filtering (DACIF) is founded based on weighted average consensus to obtain accurate state estimation. Two‐step information fusion, including the information fusion of state estimations and noise statistics, is derived to enhance the precision of state estimations. Meanwhile, a novel weighted rule is devised based on the state and measurement innovation vectors to improve the accuracy of distribution information fusion. Next, the estimation errors of DACIF are proved to be bounded in mean square. Simulations and semisimulation experiments are conducted to verify the effectiveness of the proposed algorithm.

Performing data fusion requires estimates of the state of a system to be converted to a common representation. The mean and covariance representation is the lingua franca of modern systems engineering. In particular, the covariance intersection (CI) 2 and Kalman lter (KF)3 algorithms provide mechanisms for fusing state estimates de ned in terms of means and covariances, where each mean vector de nes the nominal state of the system and its associated error covariance matrix de nes a lower bound on the squared error. However, most data fusion applications require the fusion of mean and covariance estimates de ning the state of a system in different coordinate frames. For example, a tracking system might maintain estimates in a global Cartesian coordinate frame, whereas observations of the tracked objects are generated in the local coordinate frames of various sensors. Therefore, a transformation must be applied to convert between the global coordinate frame and each local coordinate frame.

In this letter, a new type of fast covariance intersection (CI) rule to deal with unknown correlations is proposed. Different from the existing CI and its variants, our approach can obtain the optimized CI weights offline while preserving a guaranteed filtering accuracy and stability in the online implementation stage. To this end, the connection between the upper bound of the fused error covariances and the observability Gramian is first established. Next, the optimization of error covariances is converted into the optimization of observability Gramian, which is made of system matrices. Accordingly, the CI weights can be calculated prior to the real implementation. Moreover, the stability result of the fusion is also established with the help of the proposed jointly uniform observability condition. At last, simulations are given to demonstrate the effectiveness of the proposed fast CI method.

For the multi-channel ARMA signal with two sensors, by the classical Kalman filtering method and the covariance intersection (CI) fusion method, a covariance intersection fusion steady-state Kalman signal smoother is presented, which is independent of the unknown cross-covariance. It is proved that its accuracy is higher than that of each local Kalman signal smoother, and is lower than that of the optimal signal fuser weighted by matrices. The geometric interpretation of the above accuracy relations are presented based on the covariance ellipses. A simulation example result shows its effectiveness and correctness.