Robust Distributed Kalman Filter for Wireless Sensor Networks with Uncertain Communication Channels

Algorithm 1

Robust distributed fusion algorithm for node 𝑖.

Given ∑^{𝑖𝑡|𝑡−1}, error covariance, 𝑥̂^{𝑖𝑡|𝑡−1}, previous estimate, consensus

update parameter 𝜀, and the window size Δ.

 1. Obtain measurement 𝑦^𝑖𝑡=𝐶(𝛾^𝑖𝑡)𝑥_𝑡+𝜐^𝑖𝑡+𝑧^𝑖𝑡,𝑖=1,…,𝑁.

 2. For each measurement solve L1-norm optimization problem,

   reject outliers as given in (3.5) and then obtain the trimmed

   measurements: 𝑦^𝑖𝑡̂=𝑦^𝑖𝑡−𝑧^𝑖𝑡̂.

 3. Calculate the mode probability Pr(𝛾^𝑖𝑡∣𝑦̂^{𝑖𝑡−Δ∶𝑡}).

   Given Pr(𝛾^{𝑖𝑡−Δ}∣𝑦̂^{𝑖𝑡−Δ}),

   For 𝑠=𝑡−Δ∶𝑡

    Evaluate measurement likelihood for 𝑦^𝑖𝑠̂.

    Evaluate the Bayesian recursion (3.8)-(3.9).

   End

   Decide the channel mode 𝛾̂^𝑖𝑡 using threshold testing.

 4. Compute contribution term of information state and matrix

   such that

          𝑢^𝑖𝑡=(𝐶^𝑖𝑡(𝛾̂^𝑖𝑡))^𝑇(𝑅^𝑖)⁻¹𝑦^𝑖𝑡̂,

         𝑈^𝑖𝑡=(𝐶^𝑖𝑡(𝛾̂^𝑖𝑡))^𝑇(𝑅^𝑖)⁻¹𝐶^𝑖𝑡(𝛾̂^𝑖𝑡).

 5. Broadcast message 𝑚^𝑖𝑡=(𝑢^𝑖𝑡,𝑈^𝑖𝑡,𝑥̂^{𝑖𝑡|𝑡−1}) to neighbors in 𝐿_𝑖.

 6. Collect messages 𝑚^𝑟𝑡=(𝑢^𝑟𝑡,𝑈^𝑟𝑡,𝑥̂^{𝑟𝑡|𝑡−1}) from neighbors.

 7. Aggregate the information states and matrices of neighbors

   including node 𝑖: 𝐽_𝑖=𝐿_𝑖∪{𝑖}:

         𝑔^𝑖𝑡=∑_{𝑟∈𝐽𝑖}𝑢^𝑟𝑡,𝑆^𝑖𝑡=∑_{𝑟∈𝐽𝑖}𝑈^𝑟𝑡.

 8. Compute the Kalman-Consensus estimate:

          (𝑀^𝑖𝑡)⁻¹=(Φ^{𝑖𝑡∣𝑡−1})⁻¹+𝑆^𝑖𝑡,

 𝑥̂^{𝑖𝑡∣𝑡}=𝑥̂^{𝑖𝑡∣𝑡−1}+𝑀^𝑖𝑡(𝑔^𝑖𝑡−𝑆^𝑖𝑡𝑥̂^{𝑖𝑡|𝑡−1})+𝜀𝑀^𝑖𝑡1+‖𝑀^𝑖𝑡‖∑_{𝑟∈𝐽𝑖}(𝑥̂^{𝑟𝑡∣𝑡−1}−𝑥̂^{𝑖𝑡∣𝑡−1}).

   Prediction stage

Φ^{𝑖𝑡+1∣𝑡}⟵𝐴𝑀^𝑖𝑡𝐴^𝑇+𝑄,

           𝑥̂^{𝑖𝑡+1∣𝑡}⟵𝐴𝑥̂^{𝑖𝑡∣𝑡}.