Consensus algorithm for networked dynamic systems is an important research problem for data fusion in sensor networks. In this paper, the distributed filter with consensus strategies known as Kalman consensus filter and information consensus filter is investigated for state estimation of distributed sensor networks. Firstly, an in-depth comparison analysis between Kalman consensus filter and information consensus filter is given, and the result shows that the information consensus filter performs better than the Kalman consensus filter. Secondly, a novel optimization process to update the consensus weights is proposed based on the information consensus filter. Finally, some numerical simulations are given, and the experiment results show that the proposed method achieves better performance than the existing consensus filter strategies.

In recent years, there has been a surge of interests in the area of distributed sensor networks. The advantages of distributed sensor networks lie in their low processing power, cheap memory, scalable sensing features, and fault tolerance capabilities.

One of the most basic problems for distributed sensor networks is to develop distributed algorithms [

In this paper, we firstly describe the existing distributed filter with consensus strategies. Then we make an in-depth comparison between the KCF and the ICF. Based on the ICF, we propose the consensus weights optimization for better performance of the system and refer this method as

The structure of this paper is organized as follows. In Section

Consider a dynamic process with the linear time-varying model as follows:

The observations at sensor

Let the global observation vector

Then the global observation model is given by

Since observation noises of different sensors are mutually independent, we can combine

Given the collective information

We refer to

Throughout this paper, due to the importance of the node indices, we adopt a notation that is free of the time-index

We also use the update operation

Then, we get the index-free recursive equations of a

Using the matrix inversion lemma

From (

Using the matrix inversion lemma

Based on the above derivation, the recursive equations of the Kalman filter can be rewritten as

Now we define the

When the observations are distributed among the sensors, the KF can be implemented by collecting all the sensor observations at a central location, or with observation fusion by realizing that the global observation variables in (

Recall (

Let the inverse of

Now we get the following simpler form of the filter in (

Consensus strategy defines a set of rules for a team of agents to agree on specific consensus states. With these rules each agent exchanges information with its neighboring agents and finally reaches an agreement (or consensus) concerning the consensus state over time [

Consider a team of

The

Under switching interaction topologies, if there exists a finite

In order to calculate the metropolis weights in (

In this section, we discuss an alternative approach to distribute the Kalman filtering that relies on communicating state estimates between neighboring nodes and refer to it as

In local Kalman filtering, let

We now present the Kalman consensus filter (KCF). The KCF uses consensus strategy (

The local KCF is summarized in Algorithm

(1) Consensus update

(2) If new observations are taken then the Kalman consensus state estimate are computed

(3) If time for a predication step (i.e.,

The last term in (

The algorithm structure of node

In [

(1) Compute local observation vector and matrix of node

(2) Broadcast message

(3) Receive messages from all neighbors.

(4) Compute the local aggregate information vector and matrix:

(5) Compute the Kalman consensus state estimate

(6) Update the state of the Kalman consensus filter:

The KCF discussed previously applies consensus strategy on the prior estimate to the Kalman filter and improves the state estimate of KF. However, the error covariance matrix

To distribute the estimation of the global state vector,

Let the inverses of

Then we get the following simpler form of the filter in (

We now present the information consensus filter (ICF). The ICF uses consensus strategy (

Using the fused local information vector and matrix,

(1) Consensus update

(2) If new observations are taken then the information consensus estimate are computed

(3) If time for a predication step (i.e.,

Now we make a comparison between ICF and KCF based on the state estimate

The weights

The first term in the objective function

Let the linear system under consideration be represented by a second-order discrete time-varying model:

A sensor network with 20 nodes and 51 links.

Define the averaged estimation error

Figure

The averaged estimation errors of different algorithms.

Figure

The averaged consistency estimation errors of different algorithms.

Figure

The traces of the averaged estimation covariance matrices of different algorithms.

In this paper, a description about the existing distributed Filter with consensus strategies is presented, including Kalman consensus filter (KCF) and information consensus filter (ICF). In addition, an in-depth comparison between the KCF and the ICF is made. Based on the ICF, the weights optimized information consensus filter (WO-ICF) is proposed to optimize the consensus weights. Simulation shows that both ICF and WO-ICF perform better than KCF; they improve not only the state estimate but also the error covariance matrix, and the proposed WO-ICF achieves better consistency estimation performance than ICF. Compared with the existing consensus filter, our WO-ICF achieves the best performance and is closest to the optimal centralized performance.

This work is supported in part by NSFC (Natural Science Foundation of China) Projects 61202400, Natural Science Foundation of Zhejiang Q12F020078, Shaoxing Project of Science and Technology 2011A22013, and Wenzhou Project of Science and Technology H20090054, S20100029, and H20100095.