Short communicationDiffusion distributed Kalman filter over sensor networks without exchanging raw measurements
Introduction
With the advance of microprocessor technology, energy supply and communication methods, recent few decades have witnessed the popularity of multi-sensor systems. Although access of measurements is no longer the bottleneck, it does not mean that quantities of measurements will make sense without proper information fusion methods. As the most famous information fusion technology, Kalman filter (KF) based algorithms have been used dramatically, ranging from military applications, e.g. target tracking, battlefield information fusion, integrated navigation, to civil applications, e.g. smart grid, healthy monitoring, weather forecasting; recent related reviews can refer [1], [2], [3], [4].
Generally speaking, according to whether there is a fusion center (FC) or not, KF based methods can be classified into two kinds: centralized KF (CKF) and distributed KF (DKF). In CKF, every sensor sends their own measurements or individual estimations to FC to obtain global estimations. CKF is the optimal estimator due to use all the available information of the whole network. However, CKF will cause heavy communication and computation burden at FC, especially when the number of sensors is huge. Besides, the whole system will break down once the FC fails. DKF overcomes those shortcomings of CKF, which is a completely distributed method that there is no FC and every sensor (node) communicates with its neighbor sensors. Typical examples of DKF are the consensus DKF [5], [6], [7] and diffusion DKF [2], [8], [9], [10]. Comparing with the CKF, those two kinds of DKF release the communication bandwidth burdens and take advantages of the broadcaster nature of wireless communication. Due to the fact that diffusion strategy has better estimation performance than consensus strategy [8], [11], we here adopt the former one.
The seminal paper of diffusion strategy based DKF is [8], where raw measurements and intermediate estimations are exchanged among the neighborhood of every node and the final estimations are obtained through convex combination of intermediate ones. In [9], the authors adopt a simplified covariance intersection (CI) method to adjust the convex combination weights. However, both [8], [9] need exchange raw measurements, which is unsuitable for the case sensors with intermittent observations [12], [13], [14], or for the case where it is inconvenient to transmit raw measurements regularly [15], [16], [17]. In our recent work [10], every node uses their own measurements to obtain individual estimations and adopts information sharing principle to handle with individual estimation correlations. Different from [10], we here derive a new fusion method to fuse individual estimations and the convex combination weights are also obtained through a simplified CI technology similar to [9]. Simulation results show that our new algorithm has better estimation accuracy as compared with related diffusion DKF from [8], [9], [10]. To further reduce the communication burden, we give a simplified algorithm of the former, called diffusion DKF with double CI, where the fusion period of local and final estimations is larger than the sampling period to reduce communication and energy consuming. Simulation results demonstrate that this algorithm has higher estimation accuracy as compared with [10] when the fusion period increases.
The reminder of this paper is organized as follows. In Section 2 and Section 3, we present the system model, KF and diffusion DKF. Our two new algorithms are derived and analyzed in 4 Diffusion DKF with CI, 5 Diffusion DKF with double CI. Section 6 investigates the mean and the convergence performance of the proposed algorithms. In Section 7, a target tracking example is presented to demonstrate the effectiveness of the proposed algorithms, while using the algorithms from [8], [9], [10] for comparison. Conclusions are made in the final part.
Section snippets
System model and KF algorithm
Consider the following linear system:where vectors , are system state and measurements of sensor at time , , and have proper dimensions. System noise and measurement noise are uncorrelated zero mean white noise sequences, and their covariance matrixes are and . The initial state is uncorrelated with both and for all time instants and all sensors, and its mean and covariance are both
Diffusion DKF
In the diffusion strategy, every sensor only communicates with the neighboring sensors. Consider a network denoted by an undirected connected graph , where nodes and edges . We denote the neighbor of sensor as , and the number of the neighbor sensors is . The whole algorithm is initialized with and , and is presented as follows [8]:
Incremental update:
Diffusion DKF with CI
In the diffusion DKF, every sensor exchanges , and to obtain local estimations. To avoid the exchange of raw measurements, in our new algorithm, named as diffusion DKF with CI, every sensor firstly runs KF iterations based on own measurements using Eqs. (3), (4), (5), (6), (7), (8) to obtain individual estimations and . The local estimations can be obtained through the following equation [1]:
Diffusion DKF with double CI
In [26], it has been pointed out that, in sensor networks, energy is mainly consumed by communication process instead of computation. If the frequency of communication is reduced, energy can also be saved correspondingly, which is of importance in real applications where sensors have limited energy supply or inconvenient to change batteries. Due to the above reasons, we here design a novel diffusion DKF with double CI, where both the local and final estimations are periodically obtained through
Performance analysis
In this section, we will analysis the mean and convergence of proposed algorithms under conservative conditions using mathematical induction method. We firstly introduce two assumptions for preparation. Assumption 1 The system model (1)–(2) is time-invariant, which means that matrixes , , , and will not change with time, and we use , , , and in the following. Assumption 2 The pair is completely detectable and the pair is completely stabilizable.
Assumption 2 ensures the convergence
Simulation results
Here we consider a target tracking example to illustrate the performance of our proposed algorithms. The system discrete state-model is from [27]:where , , , , , , , , . When , , and when , . is the sampling period. And we set the initial values
Conclusions
In this paper, we develop two algorithms to cope with the state estimation problems over sensor networks without using raw measurements. The first algorithm yields better estimation accuracy than existing several diffusion based DKF solutions, and the second algorithm further release the burden of communication and computation through fusing at a selected rate, which yields a compromise between estimation accuracy and energy (communication) consuming. Both algorithms are unbiased and uniform
Acknowledgments
The authors would like to thank editor and anonymous reviewers for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (61201409, 61371173), and Fundamental Research Funds for the Central University of Harbin Engineering University (HEUCFQ20150407).
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