# Consensus-Based Distributed Target Tracking with False Data Injection Attacks over Radar Network
## 4 Numerical Simulation  
In order to show the effectiveness of the proposed DSE algorithm with FDI attacks, two cases are presented. In Section 4.1, assume there are two radars under FDI attacks in an eight-radar network. A comparison between the traditional hybrid consensus-based DSE method [12] (denoted as DSE) and the proposed DSE algorithm with FDI attacks (denoted as DSE-FDI) was made, which shows that DSE-FDI performed better than DSE method. In Section 4.2, the results are given for when four radars were under FDI attacks in an eight-radar network. By assuming modeling of the target dynamics by the constant velocity model [29], then the matrix can be given by  
\[\mathbf{F}(k-1)=\left[ \begin{array}{ccccc}1&0&0&\Delta t&0&0\\ 0&1&0&0&\Delta t&0\\ 0&0&1&0&0&\Delta t\\ 0&0&0&1&0\\ 0&0&0&0&1\\ \end{array} \right] \tag{50}\]  
\[\mathbf{Q}(k-1)=\sigma_{w}^{2}\left[ \begin{array}{ccccc}\Delta t^{3}/3&0&0& \Delta t^{2}/2&0&0\\ 0&\Delta t^{3}/3&0&0&\Delta t^{2}/2&0\\ 0&0&\Delta t^{3}/3&0&0&\Delta t^{2}/2\\ \Delta t^{2}/2&0&0&\Delta t&0&0\\ 0&0&\Delta t^{2}/2&0&0&\Delta t&0\\ 0&0&\Delta t^{2}/2&0&0&\Delta t\\ \end{array} \right] \tag{51}\]  
where \(\sigma_{w}\) is the standard deviation of the process noise and \(\Delta t\) is the time step. \begin{table}
\begin{tabular}{l} \hline \hline
**Algorithm 1:** \\ \hline
1: **Procedure1: Injected data detection:** \\
2: **for** each radar \(i=1,\ldots,N\)**do** \\
3: **Calculates \(I_{i}(k)\) using Equation (12). \\
4: **If \(I_{i}(k)>I_{hit}\), \(\beta_{i}=1\)** \\
5: **Else \(\beta_{i}=0\)**.** \\
6: **Procedure2: Attack parameter estimation:** \\
7: **for** each group \\
8: **Run the time update steps using Equations (20) and (21). \\
9: **Calculate pseudo-measurement \(\mathbb{Y}(k)\) the pseudo-measurement matrix \(\mathbb{H}(k)\) and the measurement noise covariance \(\mathbb{R}(k)\)** \\
10: **Run the measurement update steps using Equations (22)–(26). \\
11: **Procedure3: Time update for each group:** \\
12: **Each group carries out the time update using Equations (27)–(30). \\
13: **Procedure4: Hybrid consensus:** \\
14: **for** each group \\
15: **Calculates the information state contribution \(\hat{\mathbf{i}}_{i}(k)\) and information matrix contribution \(\mathbf{I}_{i}(k)\) using Equations (33–36). \\
16: **Calculates the predicted information matrix \(\mathbf{Y}_{i}(k|k-1)\) and information state \(\Omega_{i}(k|k-1)\) using Equations (29) and (30). \\
17: **for** \(t=1,2,\ldots,T\)**, do** \\
18: **Exchange information among the groups.** \\
19: **Emply consensus approach to the exchanged information using Equations (41)–(44) \\
20: **Each group updates its information state and information matrix using Equations (45) and (46) \\
21: **Procedure5: State estimation:** \\
22: **Each group calculate its state estimate \(\hat{\mathbf{x}}_{i}(k|k)\) and covariance \(\mathbf{P}_{i}(k|k)\) \\ \(\mathbf{P}_{i}(k|k)=\mathbf{Y}_{i}(k|k)^{-1}\) \\ \(\hat{\mathbf{x}}_{i}(k|k)=\mathbf{P}_{i}(k|k)\cdot\Omega_{i}(k|k)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The proposed DSE algorithm with FDI attacks. In the scenario, the target is tracked by an eight-radar network divided into four groups, as shown in Figure 1. The adjacency matrix of the groups is expressed as  
\[A(G)=\left[ \begin{array}{cccc}0&1&1&0\\ 1&0&0&1\\ 1&0&0&1\\ 0&1&1&0\end{array} \right] \tag{52}\]  
The true initial state of the target is \(\mathbf{x}_{0}=\) [90,000 m 10,000 m 10,000 m 40 m/s 40 m/s 40 m/s]\({}^{\rm T}\). The total simulation time is 20 s. The time step is 0.1 s. The standard deviation of the process noise \(\sigma_{w}=6\). The standard deviation of the measurement noise \(\sigma_{ir}=10\), \(\sigma_{i\theta}=\sigma_{ip}=1\times 10^{-3}\) for \(i=1,2,\ldots 8\). The attack sequence is \(r_{i}^{a}(k)\sim N(20,0.5)\), \(\theta_{i}^{a}(k)\sim N\big{(}5\times 10^{-3},0.2\times 10^{-3}\big{)}\), \(\varphi_{i}^{a}(k)\sim N\big{(}5\times 10^{-3},0.2\times 10^{-3}\big{)}\). The consensus steps of two filters are \(T=1\). The threshold \(J_{th}\) is set to be 12.838.