SCIENCE CHINA Information Sciences, Volume 63 , Issue 11 : 210202(2020) https://doi.org/10.1007/s11432-020-3000-1

Fully distributed variational Bayesian non-linear filter with unknown measurement noise in sensor networks

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  • ReceivedMay 13, 2020
  • AcceptedJul 9, 2020
  • PublishedOct 22, 2020



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    Algorithm 1 VB-DACIF implemented at node $~i~$

    Require:The posterior estimates $~\hat{\boldsymbol{x}}_{i,k-1|k-1}~$, $~\boldsymbol{P}_{i,k-1|k-1}~$, $~v_{i,k-1}~$, and $~\boldsymbol{V}_{i,k-1}~$ at time instant $~k-1~$. _ Time update:

    Compute the predicted $~\hat{\boldsymbol{x}}_{i,k|k-1}~$ and $~\boldsymbol{P}_{i,k|k-1}~$ via Eqs. (16) and (17);

    Compute the predicted $~v_{i,k|k-1}~$, $~\boldsymbol{V}_{i,k|k-1}~$ via Eq. (19);

    Fuse the predicted estimates obtained from $~j\in\mathcal{J}_i~$ via Eqs. (20)–(25);_ Recursive measurement update:

    Initialization: set $~\boldsymbol{Y}^0_{i,k|k}~=~\bar{\boldsymbol{Y}}_{i,k|k-1},\;\hat{\boldsymbol{y}}^0_{i,k|k}~=~\bar{\boldsymbol{y}}_{i,k|k-1},\;~v_{i,k}~=~v_{i,k|k-1}~+~1,\;\boldsymbol{V}^0_{i,k}~=~\boldsymbol{V}_{i,k|k-1}$;

    VB approximation;

    for $~l~=~1:L~$

    Compute the local measurement information contribution: if $~i\in\mathcal{S}~$, $~\hat{\boldsymbol\varLambda}_{i,k}^l~=~v_{i,k}\boldsymbol{V}^{l-1}_{i,k},\;~\boldsymbol{u}_{i,k}^l~=~\boldsymbol{\mathcal{H}}_{i,k}^{\text{T}}\hat{\boldsymbol\varLambda}_{i,k}(\tilde{\boldsymbol{z}}_{i,k}~-~\boldsymbol{\mathcal{H}}_{i,k}\boldsymbol{x}_{i,k|k-1}),\;~\boldsymbol{U}_{i,k}^l~=~\boldsymbol{\mathcal{H}}_{i,k}^{\text{T}}\hat{\boldsymbol\varLambda}_{i,k}\boldsymbol{\mathcal{H}}_{i,k}$; if $~i\in\mathcal{C}~$, $~\boldsymbol{u}_{i,k}^l~=~\boldsymbol{0},\;\boldsymbol{U}_{i,k}^l~=~\boldsymbol{0}~$;

    Update the local posterior estimate

  • Table 1  

    Table 1Averaged RMSEs for different algorithms

    Position (m) Velocity (m/s) Turn rate (deg/s)
    CubICF 131.473 206.281 18.004
    HCCKF 120.344 41.010 7.200
    CLCP-UKF 123.210 908.755 59.644
    VB-CCKF 64.694 34.689 3.281
    VB-DACIF 36.025 14.470 1.627
  • Table 2  

    Table 2Averaged position RMSEs (m) with different VB iterations

    Algorithm $~L=1~$ $~L=2~$ $~L=3~$ $~L=4~$ $~L=5~$
    VB-CCKF 64.694 62.015 63.198 64.271 64.871
    VB-DACIF 36.025 35.739 37.874 38.244 38.486
  • Table 3  

    Table 3Averaged velocity RMSEs (m/s) with different VB iterations

    Algorithm $~L=1~$ $~L=2~$ $~L=3~$ $~L=4~$ $~L=5~$
    VB-CCKF 34.689 34.490 34.388 35.356 36.170
    VB-DACIF 14.470 14.386 14.214 14.392 14.645
  • Table 4  

    Table 4Averaged turn rate RMSEs (deg/s) with different VB iterations

    Algorithm $~L=1~$ $~L=2~$ $~L=3~$ $~L=4~$ $~L=5~$
    VB-CCKF 3.281 3.214 3.121 3.289 3.312
    VB-DACIF 1.627 1.622 1.617 1.620 1.621
  • Table 5  

    Table 5DoEs with different VB iterations

    Algorithm $~L=1~$ $~L=2~$ $~L=3~$ $~L=4~$ $~L=5~$
    VB-CCKF 40.725 36.120 31.026 30.579 30.192
    VB-DACIF 21.066 20.310 20.086 19.984 19.927