Algorithm 2 Operation of the resilient estimation with Gaussian disturbance/noise

Input: ${\widehat{z}}_1$, ${\widehat{z}}_2$, ⋯, ${\widehat{z}}_{\mathrm{p}}$, $P_1$, $P_{12}$, ⋯, $P_{\mathrm{p}(\mathrm{p}-1)}$, $P_{\mathrm{p}}$ Output: ${\mathcal{J}}^{*}$, $\widehat{x}$, g, f Initialization: ${\mathcal{J}}^{*}=\left[\mathrm{p}\right]$, $\widehat{x}=\Psi \widehat{z}$, $g=0$, $f=0$  1: while system (1) is running do  2:    $\widehat{x}={\Psi }_{{\mathcal{J}}^{*}}{\widehat{z}}_{{\mathcal{J}}^{*}}$  3:    $r={\widehat{z}}_{{\mathcal{J}}^{*}}-{\Phi }_{{\mathcal{J}}^{*}}\widehat{x}$  4:    $\zeta ={\left(P_{{\mathcal{J}}^{*}}-{\Phi }_{{\mathcal{J}}^{*}}{\Psi }_{{\mathcal{J}}^{*}}{P}_{{\mathcal{J}}^{*}}\right)}^{-\frac{1}{2}}r$  5:    $g={\zeta }^{\top }\zeta$  6:    if $g\le {\Delta }_{TH}^{{\mathcal{J}}^{*}}$ then  7:       $f=0$  8:    else if $g>{\Delta }_{TH}^{{\mathcal{J}}^{*}}$ then  9:       $f=1$  10:       for $\mathcal{J}\subset \left[\mathrm{p}\right]$ satisfying $\left|\mathcal{J}\right|=\mathrm{p}-\mathsf{q}$ do  11:          ${\widehat{x}}^{\mathcal{J}}={\Psi }_{\mathcal{J}}{\widehat{z}}_{\mathcal{J}}$  12:          $r^{\mathcal{J}}={\widehat{z}}_{\mathcal{J}}-{\Phi }_{\mathcal{J}}{\widehat{x}}^{\mathcal{J}}$  13:          ${\zeta }^{\mathcal{J}}={\left(P_{\mathcal{J}}-{\Phi }_{\mathcal{J}}{\Psi }_{\mathcal{J}}{P}_{\mathcal{J}}\right)}^{-\frac{1}{2}}{r}^{\mathcal{J}}$  14:          $g^{\mathcal{J}}={{\zeta }^{\mathcal{J}}}^{\top }{\zeta }^{\mathcal{J}}$  15:       end for  16:       ${\displaystyle {\mathcal{J}}^{*}=\underset{\begin{array}{c}\mathcal{J}\subset \left[\mathrm{p}\right]\\ \left|\mathcal{J}\right|=\mathrm{p}-\mathsf{q}\end{array}}{arg\; max}{p}_{{\mathbf{g}}_{\mathcal{J}}}\left(g^{\mathcal{J}}\right)}$  17:    end if  18: end while