Distributed Filter with Consensus Strategies for Sensor Networks

Algorithm 2

Suboptimal Kalman consensus filter: DKF Algorithm with an estimator that has a rigorously derived consensus term (message passing during one time cycle for node).

Initialization:

        𝑥_𝑖=𝑥(0), 𝑃_𝑖=𝑃₀, and message 𝑚_𝑗={𝑦_𝑗,𝑆_𝑗,𝑥_𝑗}

While new data exists do

(1) Compute local observation vector and matrix of node 𝑖:

        𝑦_𝑖=𝐻_𝑖^𝑇𝑅_𝑖⁻¹𝑧_𝑖  𝑆_𝑖=𝐻_𝑖^𝑇𝑅_𝑖⁻¹𝐻_𝑖

(2) Broadcast message 𝑚_𝑖={𝑦_𝑖,𝑆_𝑖,𝑥_𝑖} to neighbors.

(3) Receive messages from all neighbors.

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

        𝑦^𝑖=∑_{𝑗∈𝐽𝑖}𝑦_𝑗  𝑆^𝑖==∑_{𝑗∈𝐽𝑖}𝑆_𝑗

(5) Compute the Kalman consensus state estimate

        𝑥_𝑖=𝑥_𝑖+𝑀_𝑖(𝑦^𝑖−𝑆^𝑖𝑥_𝑖)+𝛾𝑃_𝑖∑_{𝑗∈𝑁𝑖}(𝑥_𝑗−𝑥_𝑖)

        𝑀_𝑖=(𝑃_𝑖⁻¹+𝑆^𝑖)⁻¹

        𝜀𝛾=1+𝑃_𝑖,  ‖𝑋‖=tr(𝑋^𝑇𝑋)^1/2

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

        𝑃_𝑖⁺=𝐴𝑀_𝑖𝐴^𝑇+𝐵𝑄𝐵^𝑇

        𝑥_𝑖⁺=𝐴𝑥_𝑖

End While