Diffusion distributed Kalman filter over sensor networks without exchanging raw measurements

doi:10.1016/j.sigpro.2016.07.033

Signal Processing

Volume 132, March 2017, Pages 1-7

https://doi.org/10.1016/j.sigpro.2016.07.033 Get rights and content

Highlights

•
Proposed two algorithms avoid using raw measurements
•
The first one yields better estimation accuracy than existing similar solutions.
•
The second one further releases communication burden by fusing at a selected rate.
•
Both algorithms are unbiased and uniform stable under some assumptions.
•
Target tracking examples demonstrate the effectiveness of proposed algorithms.

Abstract

In this paper, we propose a new diffusion strategy based distributed state estimation algorithm over sensor networks. In the proposed algorithm, every sensor only communicates with their neighboring sensors, and only intermediate estimation information is exchanged to avoid sharing raw measurements, which may be unavailable or inconvenient to be transmitted under some circumstances. Local estimations are obtained through a new method and the convex combination weights are obtained through covariance intersection (CI) technology. To further release the communication burden and energy consuming , one simplified algorithm is also given, where the local and final estimations are fused at a selected rate. We analyze the mean and convergence performances of proposed algorithms under some assumptions. Numerical simulations show that the first algorithm has better estimation accuracy when comparing with several existing diffusion based methods, and the latter simplified algorithm has good estimation accuracy but greatly reduced communication burden and energy consuming.

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: $x_{i} = F_{i - 1} x_{i - 1} + G_{i - 1} w_{i - 1}$ $y_{k, i} = H_{k, i} x_{i} + v_{k, i}$ where vectors $x_{i} \in R^{m}$ , $y_{k, i} \in R^{s_{k}}$ are system state and measurements of sensor $k$ $(k = 1, .. ., N)$ at time $i$ , $F_{i - 1}$ , $G_{i - 1}$ and $H_{k, i}$ have proper dimensions. System noise $w_{i}$ and measurement noise $v_{k, i}$ are uncorrelated zero mean white noise sequences, and their covariance matrixes are $Q_{i} > 0$ and $R_{k, i} > 0$ . The initial state $x_{0}$ is uncorrelated with both $w_{i}$ and $v_{k, i}$ 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 $G = (V, E)$ , where nodes $V = {1, 2, .. ., N}$ and edges $E \subseteq V \times V$ . We denote the neighbor of sensor $k$ as $N_{k} = {j \in V : (k, j) \in E} \cup {k}$ , and the number of the neighbor sensors is $n_{k}$ . The whole algorithm is initialized with ${\hat{x}}_{k, 0 | - 1}^{l o c} = 0$ and $P_{k, 0 | - 1}^{l o c} = Π_{0}$ , and is presented as follows [8]:

Incremental update: ${\hat{x}}_{k, i | i}^{l o c} = {\hat{x}}_{k, i | i - 1}^{l o c} + P_{k, i | i}^{l o c} (\sum_{l \in N_{k}} H_{l, i}^{T} R_{l, i}^{- 1} y_{l, i} - \sum_{l \in N_{k}} H_{l, i}^{T} R_{l, i}^{-}$

Diffusion DKF with CI

In the diffusion DKF, every sensor exchanges $y_{l, i}$ , $R_{l, i}$ and $H_{l, i}$ 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 ${\hat{x}}_{k, i | i}^{ind}$ and ${\hat{x}}_{k, i | i - 1}^{ind}$ . The local estimations can be obtained through the following equation [1]: ${\hat{x}}_{k, i | i}^{l o c} = P_{k, i | i}^{l o c} [{(P_{k, i | i - 1}^{l o c})}^{- 1} {\hat{x}}_{k, i | i - 1}^{l o c} + \sum_{l \in N_{k}} ({(P_{l, i | i}^{i n d})}^{- 1}$

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 $F_{i}$ , $G_{i}$ , $H_{k, i}$ , $R_{k, i}$ and $Q_{i}$ will not change with time, and we use $F$ , $G$ , $H_{k}$ , $R_{k}$ and $Q$ in the following.

Assumption 2

The pair ${F, H_{k}}$ is completely detectable and the pair ${F, G Q^{\frac{1}{2}}}$ 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]: $x_{i} = F x_{i - 1} + G w_{i - 1} y_{k, i} = H_{k} x_{i} + v_{k, i} k = 1, .. ., 20$ where $F = [\begin{matrix} 1 & T & T^{2} \\ 0 & 1 & T \\ 0 & 0 & 1 \end{matrix}]$ , $G = [\begin{matrix} 0.5 T^{2} \\ T \\ 1 \end{matrix}]$ , $Q = 9$ , $R_{1, 7, 13, 19} = diag (2, 2.5)$ , $R_{2, 8, 14, 20} = diag (1.5, 4)$ , $R_{3, 9, 15} = diag (4.5, 2.5)$ , $R_{4, 10, 16} = diag (1.5, 3.5)$ , $R_{5, 11, 17} = diag (4.5, 4)$ , $R_{6, 12, 18} = diag (3, 2.5)$ . When $k = 1, 3, .. ., 19$ , $H_{k} = [\begin{matrix} 1 & 1 & 0 \\ 0 & 1 & 1 \end{matrix}]$ , and when $k = 2, 4, .. ., 20$ , $H_{k} = [\begin{matrix} 0 & 1 & 1 \\ 1 & 1 & 0 \end{matrix}]$ . $T = 2s$ is the sampling period. And we set the initial values ${\hat{x}}_{k, 0 | - 1} = 0$

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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Short communicationDiffusion distributed Kalman filter over sensor networks without exchanging raw measurements

Highlights

Abstract

Introduction

Section snippets

System model and KF algorithm

Diffusion DKF

Diffusion DKF with CI

Diffusion DKF with double CI

Performance analysis

Simulation results

Conclusions

Acknowledgments

Signal Process.

J. Frankl. Inst.

Automatica

Signal Process.

Automatica

Automatica

Inf. Fusion

Inf. Sci.

Distributed Kalman filtering: a bibliographic review

IET Control Theory Appl.

Adaptive networks

Proc. IEEE

Globally optimal distributed Kalman filtering fusion

Sci. China Inf. Sci.

Multisensor fusion and integration: theories, applications, and its perspectives

IEEE Sens. J.

Short communication
Diffusion distributed Kalman filter over sensor networks without exchanging raw measurements