# Consensus-Based Distributed Target Tracking with False Data Injection Attacks over Radar Network
## 3 Distributed State Estimation Algorithm with False Data Injection Attacks
### Attack Parameter Estimation  
Very few papers focus on how to estimate the unknown and time-varying attack parameters and eliminate its effect on tracking performance. Inspired by the sensor bias estimation method proposed in [25], we propose a simple-yet-effective attack parameter estimation solution based on a pseudo-measurement equation. Every two radars in the network are seen as a group. Taking the radar \(m\),\(n\) group as an example, assuming that the radar \(m\) is the leader of the group, the detailed derivation is given as follows.  
Define the pseudo-measurement as  
\[\mathbb{Y}(k)=\mathbf{y}_{m}(k)-\mathbf{y}_{n}(k) \tag{13}\]  
where \(\mathbf{y}_{m}(k)\) and \(\mathbf{y}_{n}(k)\) are the measurements of radars \(m\) and \(n\), respectively.  
If radar \(m\) is attacked, substitute Equation (8) into Equation (13), and we can obtain the pseudo-measurement equation of the attack parameter:  
\[\mathbb{Y}(k)=\mathbb{H}(k)\mathbb{A}(k)+\mathbb{V}(k) \tag{14}\]  
where the pseudo-measurement matrix is expressed as  
\[\mathbb{H}(k)=\mathbf{A}_{m}(k) \tag{15}\]  
The attack parameter vector is  
\[\mathbb{A}(k)=\mathbf{a}_{m}(k) \tag{16}\]  
The pseudo-measurement noise is expressed as  
\[\mathbb{V}(k)=\mathbf{v}_{m}(k)-\mathbf{v}_{n}(k) \tag{17}\]  
Note that the measurement noise \(\mathbf{v}_{m}(k)\) and \(\mathbf{v}_{n}(k)\) are zero-mean Gaussian noise. Thus, the pseudo-measurement noise covariance is  
\[\mathbb{R}(k)=\mathbf{R}_{m}(k)+\mathbf{R}_{n}(k) \tag{18}\]  
The time-varying attack parameter can be modeled as  
\[\mathbb{A}(k)=\mathbb{F}(k-1)\mathbb{A}(k-1)+\mathbb{W}(k-1) \tag{19}\]  
where \(\mathbb{F}(k-1)\) is the transition matrix and \(\mathbb{W}(k-1)\) is the process noise of the attack parameter with a zero mean and a variance of \(\mathbf{Q}_{\mathbf{a}}(k-1)\).  
Based on the linear pseudo-measurement equation, the standard Kalman filter can be used to obtain the estimation of the attack parameter. Assume that the attack parameter estimate is \(\mathbb{\hat{A}}(k-1|k-1\,)\) at time \(k-1\) and its corresponding covariance \(\mathbf{P}_{\mathbf{a}}(k-1|k-1\,)\) is available. The attack parameter can be recursively estimated as follows.  
Calculate the predicted attack parameter based on Equation (19):  
\[\mathbb{\hat{A}}(k|k-1\,)=\mathbb{F}(k-1)\mathbb{\hat{A}}(k-1|k-1\,) \tag{20}\]  
Calculate the predicted covariance as follows:  
\[\mathbf{P}_{\mathbf{a}}(k|k-1\,)=\mathbb{F}(k-1)\mathbf{P}_{\mathbf{a}}(k-1|k-1\,)\mathbb{F}^ {T}(k-1)+\mathbf{Q}_{\mathbf{a}}(k-1) \tag{21}\]  
Obtain the measurements from both radars, and calculate the pseudo-measurement \(\mathbb{Y}(k)\), the pseudo-measurement matrix \(\mathbb{H}(k)\) and the measurement noise covariance \(\mathbb{R}(k)\) according to Equations (13), (15) and (18), respectively.  
Calculate the predicted measurement based on Equation (14):  
\[\mathbb{\hat{Y}}(k)=\mathbb{H}(k)\mathbb{\hat{A}}(k|k-1\,) \tag{22}\]  
Calculate the gain as follows:  
\[\mathbf{S}(k)=\mathbb{R}(k)+\mathbb{H}(k)\mathbf{P}_{\mathbf{a}}(k|k-1\,)\mathbb{H}^{T}(k) \tag{23}\]\[\mathbf{K}(k)=\mathbf{P}_{a}(k|k-1)\,\mathbb{H}^{T}(k)\mathbf{S}^{-1}(k) \tag{24}\]  
Update the attack parameter estimation and the associated covariance to be  
\[\hat{\mathbb{A}}(k|k)=\hat{\mathbb{A}}(k|k-1)+\mathbf{K}(k)\Big{(}\mathbb{Y}(k )-\hat{\mathbb{Y}}(k)\Big{)} \tag{25}\]  
\[\mathbf{P}_{a}(k|k)=\mathbf{P}_{a}(k|k-1)-\mathbf{K}(k)\mathbf{S}(k)\mathbf{K} ^{T}(k) \tag{26}\]  
**Remark 1**.: _The above derivation is based on the assumption that radar \(m\) is under FDI attacks. If the detection result shows that there is no radar under attack, then \(\hat{\mathbb{A}}(k|k)\) is a zero vector._