# Consensus-Based Distributed Target Tracking with False Data Injection Attacks over Radar Network
## 2 Problem Formulation
### Target Dynamics and Radar Measurement Model  
Consider that the target dynamic is modeled as follows:  
\[x(k)=\textbf{F}(k-1)\textbf{x}(k-1)+\textbf{w}(k-1) \tag{2}\]  
where \(x(k)\in\mathbb{R}^{n}\) denotes the state at time \(k,\textbf{x}=\left[ \begin{array}{cccccc}x&y&z&\hat{x}&\dot{y}&\dot{z}\end{array}  \right]^{T}\); \(F\) is the transition matrix; \(\textbf{w}(k-1)\) is the zero-mean Gaussian process noise; and the process noise covariance is \(\textbf{Q}(k-1)\). We consider the ground-based active radars used to track the target, where the measured values are generally in spherical coordinates. Therefore, the sensing model of the \(i\)th radar under an FDI attack is described in spherical coordinates as follows [25; 26]:  
\[\textbf{y}_{i}^{s}(k)=\left[ \begin{array}{c}\widetilde{r}_{i}(k)\\ \widetilde{\theta}_{i}(k)\\ \widetilde{\varphi}_{i}(k)\end{array} \right]=\left[ \begin{array}{c}r_{i}(k) +r_{i}^{\sharp}(k)+v_{i}^{\sharp}(k)\\ \theta_{i}(k)+\theta_{i}^{\sharp}(k)+v_{i}^{\sharp}(k)\\ \varphi_{i}(k)+\varphi_{i}^{\sharp}(k)+v_{i}^{\sharp}(k)\end{array} \right] \tag{3}\]  
where \(\widetilde{r}_{i}(k),\widetilde{\theta}_{i}(k)\) and \(\widetilde{\varphi}_{i}(k)\) are the range, elevation and azimuth measurements in spherical coordinates, respectively, and \(r_{i}(k)\), \(\theta_{i}(k)\) and \(\varphi_{i}(k)\) are true measurements, which are simulated as follows (subscript \(i\) is omitted for simplicity):  
\[\left[ \begin{array}{c}r(k)\\ \theta(k)\\ \varphi(k)\end{array} \right]=\left[ \begin{array}{c}\sqrt{x_{r}^{2}+y_{r}^{2 }+z_{r}^{2}}\\ \arcsin\frac{z_{r}}{d}\\ \arctan 2(y_{r},x_{r})\end{array} \right] \tag{4}\]where \(\left[x_{r},y_{r},z_{r}\right]^{T}\) is the position of the target in the sensor coordinate system and  
\[\left[ \begin{array}{c}x_{r}\\ y_{r}\\ z_{r}\end{array} \right]=\mathbf{C}_{i}^{r}\left[ \begin{array}{c}x-x_{s}\\ y-y_{s}\\ z-z_{s}\end{array} \right] \tag{5}\]  
where \(\mathbf{C}_{i}^{r}\) is the coordinate transform matrix from the Earth-centered inertial coordinate system to the sensor coordinate system and \(\left[x_{s},y_{s},z_{s}\right]^{T}\) is the position of the sensor in the Earth-centered inertial coordinate system. In Equation (3), \(r_{i}^{a}(k)\), \(\theta_{i}^{a}(k)\) and \(\varphi_{i}^{a}(k)\) are unknown attack variables designed by the attacker, and \(v_{i}^{r}(k)\), \(\varphi_{i}^{\theta}(k)\) and \(v_{i}^{\varphi}(k)\) are zero-mean measurement noises in the range, elevation and azimuth with corresponding variances \(\sigma_{\mu}^{2}\), \(\sigma_{i\theta}^{2}\) and \(\sigma_{i\varphi}^{2}\), respectively. The measurement noises are assumed to be mutually independent. There are three main attack methods for false data injection: (1) the attack variable is a time-varying signal and converges to a constant value gradually; (2) the attack variable is a Gaussian random process; and (3) the attack variable is an arbitrary random variable. In practical application, the second attack is easy to implement and the most widely used. Therefore, in this chapter, the attack variable is modeled as a Gaussian random process, and the attack variable is assumed to be irrelevant to the measurement noise and the state. The false data injection variable is as follows:  
\[\mathbf{a}(k+1)=\mathbf{a}(k)+\mathbf{\omega}(k) \tag{6}\]  
where \(\mathbf{a}(k)\) denotes the attack variable vector and \(\mathbf{\omega}(k)\) is the zero-mean Gaussian noise. Since the motion equations of the targets are better modeled in Cartesian coordinates [25], we converted the 3D spherical measurements in Equation (3) into Cartesian coordinates. By transforming the measurements into Cartesian coordinates, we could perform the target state estimation within a completely linear framework. Denote \(\mathbf{y}_{i}(k)=\left[ \begin{array}{cc}\widetilde{x}_{i}(k)&\widetilde{y}_{i} (k)&\widetilde{z}_{i}(k)\end{array} \right]^{T}\) as the measurements in Cartesian coordinates. The measurement model can be written as  
\[