Distributed Filter with Consensus Strategies for Sensor Networks

Algorithm 1

Kalman consensus filter.

Initialization (for node 𝑖):

        𝑥_𝑖=𝑥(0)  𝑃_𝑖=𝑃₀

        𝜏=1     𝜏_𝑝=𝜏+𝑇_𝑝

Loop  {Local iteration on node 𝑖}

(1) Consensus update

        𝑥_𝑖^𝑐=𝑥_𝑖+∑_{𝑗∈𝑁𝑖}𝛽_𝑖𝑗[𝜏](𝑥_𝑗−𝑥_𝑖)

        𝑦^𝑖=∑_{𝑗∈𝐽𝑖}𝐻_𝑗^𝑇𝑅_𝑗⁻¹𝑧_𝑗 𝑆^𝑖=∑_{𝑗∈𝐽𝑖}𝐻_𝑗^𝑇𝑅_𝑗⁻¹𝐻_𝑗 𝜏←𝜏+1

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

   𝑥_𝑖=𝑥_𝑖^𝑐+𝑀_𝑖𝑦^𝑖−𝑆^𝑖𝑥_𝑖^𝑐=𝑥_𝑖+𝑀_𝑖𝑦^𝑖−𝑆^𝑖𝑥_𝑖+𝐼−𝑀_𝑖𝑆^𝑖∑_{𝑗∈𝑁𝑖}𝛽_𝑖𝑗[𝜏](𝑥_𝑗−𝑥_𝑖)

        𝑀_𝑖=(𝑃_𝑖⁻¹+𝑆^𝑖)⁻¹

(3) If time for a predication step (i.e., 𝜏=𝜏_𝑝) then prediction step

        𝑃_𝑖⁺=𝐴𝑀_𝑖𝐴^𝑇+𝐵𝑄𝐵^𝑇

        𝑥_𝑖⁺=𝐴𝑥_𝑖

        𝜏_𝑝=𝜏+𝑇_𝑝

End loop