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A distributed Kalman filter with

A distributed Kalman filter with

作者: 渡猫 | 来源:发表于2018-11-07 09:16 被阅读0次

\mathbf{Table\;1}
\text{Algorithm 1 - distributed Kalman filtering with network attacked}

\mathbf{Initialization:}
\quad \text{let the local prior information pair }(\Omega_{0|-1}^i,q_{0|-1}^i)=(0,0),a_0^i=0
\text{At each time k = 0,1,..., for each node i}
\quad \mathbf{(1)\;Correction:}
\qquad \text{if }i\in\mathcal{S} \text{, collect measurement }y_k^i
\qquad\quad \hat{x}_{k|k-1}^i=(\Omega_{k|k-1}^i)^{-1}q_{k|k-1}^i
\qquad\quad a_k^i=1-a_k^i\text{ if }\lVert y_{k}-\hat{x}_{k|k-1}^i\rVert<\alpha
\qquad\quad \begin{equation} q_{k|k}^i=\begin{cases} q_{k|k-1}^i+(C^i)^T(R^i)^{-1}y_k^i,&\text{if } a_{k-1}=0\\ q_{k|k-1}^i+(C^i)^T(R^i+S)^{-1}y_k^i,&\text{if } a_{k-1}=1 \end{cases} \end{equation}
\qquad\quad \begin{equation} \Omega_{k|k}^i=\begin{cases} \Omega_{k|k-1}^i+(C^i)^T(R^i)^{-1}C^i,&\text{if } a_{k-1}=0\\ \Omega_{k|k-1}^i+(C^i)^T(R^i+S)^{-1}C^i,&\text{if } a_{k-1}=1 \end{cases} \end{equation}
\qquad \text{if }i\in\mathcal{C}
\qquad\quad \left(\Omega_{k|k}^i,q_{k|k}^i\right)=\left(\Omega_{k|k-1}^i,q_{k|k-1}^i\right)
\quad \mathbf{(2)\;Information\;exchange:}
\qquad\quad \begin{equation} c_k^i=\begin{cases} 0,&\text{if } a_k^i=1\text{ and }\frac{1}{1+\beta}\Omega_{k|k}^i\leq\bar{\Omega}^i_k\leq(1+\delta)\Omega_{k|k}^i\\ 1,&\text{otherwise} \end{cases} \end{equation}
\qquad \text{if }c_k^i=1\text{ transmit }(\Omega_{k|k}^i,q_{k|k}^i)\text{ to the out-neighbors, receive }(\Omega_{k|k}^j,q_{k|k}^j)
\qquad \text{from all the in-neighbors } j\in\mathcal{N}_I\text{ for which }c_k^i=1
\quad \mathbf{(3)\;Information\;fusion:}
\qquad\quad q_{k|k}^{i,F}=\pi_{i,j}q_{k|k}^i+\sum\nolimits_{j\in\mathcal{N}_i}\pi_{i,j}\left[c_k^jq_{k|k}^j+(1-c_k^j)\frac{1}{1+\delta}\bar{q}_k^j\right]
\qquad\quad \Omega_{k|k}^{i,F}=\pi_{i,j}\Omega_{k|k}^i+\sum\nolimits_{j\in\mathcal{N}_i}\pi_{i,j}\left[c_k^j\Omega_{k|k}^j+(1-c_k^j)\frac{1}{1+\delta}\bar{\Omega}_k^j\right]
\quad \mathbf{(4)\;Prediction:}
\qquad\quad q_{k+1|k}^i=\Omega_{k+1|k}^{i}A\left(\Omega_{k|k}^{i,F}\right)^{-1}q_{k|k}^{i,F}
\qquad\quad \Omega_{k+1|k}^i=Q^{-1}-Q^{-1}A\left(\Omega_{k|k}^{i,F}+A^TQ^{-1}A\right)^{-1}A^TQ^{-1}
\qquad j\in\mathcal{N}_i+i
\qquad\quad \bar{q}_{k+1}^j=\bar{\Omega}_{k+1}^{j}A\left(\check{\Omega}_{k}^{j}\right)^{-1}\check{q}_{k}^{j}
\qquad\quad \bar{\Omega}_{k+1}^j=Q^{-1}-Q^{-1}A\left(\check{\Omega}_{k}^{j}+A^TQ^{-1}A\right)^{-1}A^TQ^{-1}
\qquad \text{where } \check{q}_{k}^{j}=c_k^jq_{k|k}^j+(1-c_k^j)\bar{q}_k^j\text{ and }\check{\Omega}_{k}^{j}=c_k^j\Omega_{k|k}^j+(1-c_k^j)\bar{\Omega}_k^j

(\Omega_{k|k}^i,
q^i_{k|k}=q^i_{k|k-1}+C^i_kR^{-1}C^i_k,\hat{x}_{0|-1}^i=(\Omega_{0|-1}^i)^{-1}q_{0|-1}^i
\quad \left|\left| 3 +\frac{7x+5}{1+y^2} \right|\right|asdas
\tilde{A}
asdasd
asd

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