# Consensus-Based Distributed Target Tracking with False Data Injection Attacks over Radar Network
## 3 Distributed State Estimation Algorithm with False Data Injection Attacks
### Hybrid Consensus  
For DSE, consensus estimation is an effective way for state estimation in a radar network that is not fully connected to achieve global consistency for all radars. Existing consensus methods, including the CE, CM, CI and HCMCI methods, were proposed and applied in [9; 10; 11; 12; 13; 14; 15; 16]. However, the above methods do not take into account the occurrence of FDI attacks. To address this problem, a novel hybrid consensus method, which can mitigate the FDI attacks and achieve global consistency in a timely manner, is proposed. In the proposed hybrid consensus procedure, each group is treated as a whole part. If the detection results show that there is no radar under attack in a group, we call the group as not being under attack; otherwise, we call the group as being under attack. Denote \(\zeta_{i}\) as the exchanged information of group \(i\). Each group receives iterative exchange information with its neighbors and updates its own information as follows:  
\[\zeta_{i}(t+1)=\sum_{s\in E_{i}}\pi^{sj}\zeta_{s}(t) \tag{31}\]where \(t\) denotes the iteration number and \(\pi^{s,i}\) denotes the consensus weight that is given by the Metropolis weights [28] as  
\[\pi^{s,i}=\left\{ \begin{array}{c}(1+\max\_d)^{-1}s\in E_{i}ands\
eq i\\ 1-\sum\limits_{\begin{array}{c}s\in E_{i}\\ s\in E_{i}\\ s\
eq i\\ 0\end{array}}\pi^{s,i}s=i\\ 0\text{\emph{otherwise}}\end{array} \right. \tag{32}\]  
where \(\max\_d=\max(d_{s},d_{i})\). The novel hybrid consensus method combines the CM and CI methods. The CM method employs the consensus method, namely Equations (31) and (32), for the exchanged \(\mathbf{i}(k)\) and \(\mathbf{I}(k)\). The \(t\)th iteration of consensus for the CM method can be described as follows:  
1. Calculate the information state contribution \(\mathbf{i}_{i}(k)\) and information matrix contribution \(\mathbf{I}_{i}(k)\).The group under an FDI attack can be expressed as \[\mathbf{i}_{i}(k)=\mathbf{H}^{T}(k)\mathbf{R}_{i}^{-1}(k)\Big{(}\mathbb{Y}_{i}(k)-\mathbf{A}_ {i}(k)\hat{\mathbb{A}}_{i}(k|k)\Big{)}\] (33) \[\mathbf{I}_{i}(k)=\mathbf{H}^{T}(k)\mathbf{R}_{i}^{-1}(k)\mathbf{H}(k)\] (34) The group which is not under an FDI attack can be expressed as \[\mathbf{i}_{i}(k)=\mathbf{H}^{T}(k)\mathbf{R}_{i}^{-1}(k)\mathbb{Y}_{i}(k)\] (35) \[\mathbf{I}_{i}(k)=\mathbf{H}^{T}(k)\mathbf{R}_{i}^{-1}(k)\mathbf{H}(k)\] (36)
2. Exchange the information contributions. For example, group \(i\) broadcasts \(\mathbf{i}_{i}(k)\) and \(\mathbf{I}_{i}(k)\) to its neighbors and receives \(\mathbf{i}_{s}(k)\) and \(\mathbf{I}_{s}(k)\) from its neighbors \(s\in E_{i}\). 3. Measurement consensus can be achieved by \[ \begin{array}{c}\mathbf{i}_{i}^{t+1}(k)=\sum\limits_{s\in E_{i}}\pi^{s,i}\mathbf{i }_{s}^{t}(k)\\ \mathbf{I}_{i}^{t+1}(k)=\sum\limits_{s\in E_{i}}\pi^{s,i}\mathbf{I}_{s}^{t}(k)\end{array} \] (37) After \(T\) iterations, the information state and information matrix can be updated as \[\Omega_{i}(k|k)=\Omega_{i}(k|k-1)+\rho_{i}(k)\mathbf{i}_{i}^{T}(k)\] (38) \[\mathbf{Y}_{i}(k|k)=\mathbf{Y}_{i}(k|k-1)+\rho_{i}(k)\mathbf{I}_{i}^{T}(k)\] (39) where \(\rho_{i}(k)\) are the scalar weights. Similarly, the CI method employs the consensus method to \(\Omega(k|k)\) and \(\mathbf{Y}(k|k)\). The \(t\)th iteration of consensus for the CI method is given as follows:  
1. Calculate the updated information matrix \(\mathbf{Y}_{i}(k|k)\) and information state \(\Omega_{i}(k|k)\) based on the predicted information matrix and information state: \[\Omega_{i}(k|k)=\Omega_{i}(k|k-1)+\mathbf{i}_{i}(k)\] (40) \[\mathbf{Y}_{i}(k|k)=\mathbf{Y}_{i}(k|k-1)+\mathbf{I}_{i}(k)\] (41) Note that for the group under an FDI attack, \(\mathbf{i}_{i}(k)\) is calculated based on Equation (33). For the group which is not under an FDI attack, \(\mathbf{i}_{i}(k)\) is calculated based on Equation (35). 2. Exchange information contributions. Each group broadcasts \(\Omega_{i}(k|k)\) and \(\mathbf{Y}_{i}(k|k)\) to its neighbors and receives information \(\Omega_{s}(k|k)\) and \(\mathbf{Y}_{s}(k|k)\), where \(s\in E_{i}\). 3.