Algorithm 2  Distributed robust Kalman filter with non-uniform local tolerance at time t.

Input:${\widehat{x}}_{k,t},V_{k,t},y_{k,t},W={\left[w_{lk}\right]}_{lk}$ with $k=1\dots N$ Output:${\widehat{x}}_{k,t+1},V_{k,t+1}$ with $k=1\dots N$ Tolerance update. Using the nominal global model, the central unit computes for every node k: $$c_{k,t}=\frac{1}{2}\left[-log\left|{\tilde{K}}_{z_{k,t}}\right|+log\left|K_{z_{k,t}}\right|+tr\left({\tilde{K}}_{z_{k,t}}{K}_{z_{k,t}}^{-1}\right)-(n+p_k)\right]$$ (31) Incremental step. Compute at every node k: $$\begin{array}{cc}\hfill {\psi }_{k,t+1}& =A{\widehat{x}}_{k,t}+A{\left(V_{k,t}^{-1}+S_k\right)}^{-1}\sum _{l\in {\mathcal{N}}_k}{C}_l^T{R}_l^{-1}\left(y_{l,t}-C_l{\widehat{x}}_{k,t}\right)+r_t\hfill \\ \hfill P_{k,t+1}& =A{\left(V_{k,t}^{-1}+S_k\right)}^{-1}{A}^T+B{B}^T\hfill \\ \hfill \mathrm{Find}& {\theta }_{k,t}\mathrm{s}.\mathrm{t}.\gamma \left(P_{k,t+1},{\theta }_{k,t}\right)=c_{k,t}\hfill \\ \hfill V_{k,t+1}& ={\left(P_{k,t+1}^{-1}-{\theta }_{k,t}I\right)}^{-1}\hfill \end{array}$$ (32) Diffusion step. Compute at every node k: $${\widehat{x}}_{k,t+1}=\sum _{l\in {\mathcal{N}}_k}{w}_{lk}{\psi }_{l,t+1}$$ (33)